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<title>Concave function</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Concave function</span></span>

</h1>

</header>

</p><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>concave function</b> is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the <a href="Hypograph_(mathematics)" title="Hypograph (mathematics)">hypograph</a> is convex. The class of concave functions is in a sense the opposite of the class of <a href="Convex_function" title="Convex function">convex functions</a>. A concave function is also <a href="Synonym" title="Synonym">synonymously</a> called <b>concave downwards</b>, <b>concave down</b>, <b>convex upwards</b>, <b>convex cap</b>, or <b>upper convex</b>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>A real-valued <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> on an <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> (or, more generally, a <a href="Convex_set" title="Convex set">convex set</a> in <a href="Vector_space" title="Vector space">vector space</a>) is said to be <i>concave</i> if, for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> in the interval and for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in [0,1]}</annotation>

</semantics>

</math></span><img src="./daf3c62599ea71319c85f715c9e590d2bab2d036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.981ex; height:2.843ex;" alt="{\displaystyle \alpha \in [0,1]}" loading="lazy"></span>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)}</annotation>

</semantics>

</math></span><img src="./6f90ff17a492a022ca98d045d9975893e524f66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.588ex; height:2.843ex;" alt="{\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)}" loading="lazy"></span></dd></dl>

<p>A function is called <i>strictly concave</i> if

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)}</annotation>

</semantics>

</math></span><img src="./2134c12763e185ad5623707195ac4584a244a3d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.588ex; height:2.843ex;" alt="{\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in (0,1)}</annotation>

</semantics>

</math></span><img src="./5df576f7940384416d1553ab063704d37bf99420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle \alpha \in (0,1)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq y}</annotation>

</semantics>

</math></span><img src="./f51b711ca7f932963cdb268b0817dc72d6258733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.676ex;" alt="{\displaystyle x\neq y}" loading="lazy"></span>.

</p><p>For a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"></span>, this second definition merely states that for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> strictly between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>, the point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z,f(z))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (z,f(z))}</annotation>

</semantics>

</math></span><img src="./d4e5db85294ef6ab432fb1b1476539f098bf2a79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.107ex; height:2.843ex;" alt="{\displaystyle (z,f(z))}" loading="lazy"></span> on the graph of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is above the straight line joining the points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,f(x))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,f(x))}</annotation>

</semantics>

</math></span><img src="./b21dd0c5c5815bc0516f679f631fd588ceb458d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.59ex; height:2.843ex;" alt="{\displaystyle (x,f(x))}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,f(y))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y,f(y))}</annotation>

</semantics>

</math></span><img src="./124083983ac72a4e1df3b5d131f19dbe76e35790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.242ex; height:2.843ex;" alt="{\displaystyle (y,f(y))}" loading="lazy"></span>.

</p><p><span class="mw-default-size" typeof="mw:File"></span>

</p><p>A function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is <a href="Quasiconvex_function" title="Quasiconvex function">quasiconcave</a> if the upper contour sets of the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(a)=\{x:f(x)\geq a\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(a)=\{x:f(x)\geq a\}}</annotation>

</semantics>

</math></span><img src="./641fe11c935936d752d85885ebf65662734aff51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.974ex; height:2.843ex;" alt="{\displaystyle S(a)=\{x:f(x)\geq a\}}" loading="lazy"></span> are convex sets.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Functions_of_a_single_variable">Functions of a single variable</h3></div>

<ol><li>A <a href="Differentiable_function" title="Differentiable function">differentiable function</a> <span class="texhtml mvar" style="font-style:italic;">f</span> is (strictly) concave on an <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> if and only if its <a href="Derivative" title="Derivative">derivative</a> function <span class="texhtml mvar" style="font-style:italic;">f ′</span> is (strictly) <a href="Monotonically_decreasing" class="mw-redirect" title="Monotonically decreasing">monotonically decreasing</a> on that interval, that is, a concave function has a non-increasing (decreasing) <a href="Slope" title="Slope">slope</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Point_(geometry)" title="Point (geometry)">Points</a> where concavity changes (between concave and <a href="Convex_function" title="Convex function">convex</a>) are <a href="Inflection_point" title="Inflection point">inflection points</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li>

<li>If <span class="texhtml mvar" style="font-style:italic;">f</span> is twice-<a href="Differentiable_function" title="Differentiable function">differentiable</a>, then <span class="texhtml mvar" style="font-style:italic;">f</span> is concave <a href="If_and_only_if" title="If and only if">if and only if</a> <span class="texhtml mvar" style="font-style:italic;">f ′′</span> is <a href="Non-positive" class="mw-redirect" title="Non-positive">non-positive</a> (or, informally, if the "<a href="Acceleration" title="Acceleration">acceleration</a>" is non-positive). If <span class="texhtml mvar" style="font-style:italic;">f ′′</span> is <a href="Negative_numbers" class="mw-redirect" title="Negative numbers">negative</a> then <span class="texhtml mvar" style="font-style:italic;">f</span> is strictly concave, but the converse is not true, as shown by <span class="texhtml"><i>f</i>(<i>x</i>) = −<i>x</i><sup>4</sup></span>.</li>

<li>If <span class="texhtml mvar" style="font-style:italic;">f</span> is concave and differentiable, then it is bounded above by its first-order <a href="Taylor_approximation" class="mw-redirect" title="Taylor approximation">Taylor approximation</a>:<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)\leq f(x)+f'(x)[y-x]}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(y)\leq f(x)+f'(x)[y-x]}</annotation>

</semantics>

</math></span></span></li>

<li>A <a href="Lebesgue_measurable_function" class="mw-redirect" title="Lebesgue measurable function">Lebesgue measurable function</a> on an interval <span class="texhtml"><b>C</b></span> is concave <a href="If_and_only_if" title="If and only if">if and only if</a> it is midpoint concave, that is, for any <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> in <span class="texhtml"><b>C</b></span> <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left({\frac {x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}}">

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left({\frac {x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}}</annotation>

</semantics>

</math></span></span></li>

<li>If a function <span class="texhtml mvar" style="font-style:italic;">f</span> is concave, and <span class="texhtml"><i>f</i>(0) ≥ 0</span>, then <span class="texhtml mvar" style="font-style:italic;">f</span> is <a href="Subadditivity" title="Subadditivity">subadditive</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation>

</semantics>

</math></span><img src="./8dc2d914c2df66bc0f7893bfb8da36766650fe47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}" loading="lazy"></span>. Proof:

<ul><li>Since <span class="texhtml mvar" style="font-style:italic;">f</span> is concave and <span class="texhtml">1 ≥ t ≥ 0</span>, letting <span class="texhtml"><i>y</i> = 0</span> we have <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(tx)=f(tx+(1-t)\cdot 0)\geq tf(x)+(1-t)f(0)\geq tf(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(tx)=f(tx+(1-t)\cdot 0)\geq tf(x)+(1-t)f(0)\geq tf(x).}</annotation>

</semantics>

</math></span></span></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b\in [0,\infty )}</annotation>

</semantics>

</math></span><img src="./14c111a21f614fb7b5a0ae3eb3ca6ba73aa58ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.174ex; height:2.843ex;" alt="{\displaystyle a,b\in [0,\infty )}" loading="lazy"></span>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)+f(b)=f\left((a+b){\frac {a}{a+b}}\right)+f\left((a+b){\frac {b}{a+b}}\right)\geq {\frac {a}{a+b}}f(a+b)+{\frac {b}{a+b}}f(a+b)=f(a+b)}">

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)+f(b)=f\left((a+b){\frac {a}{a+b}}\right)+f\left((a+b){\frac {b}{a+b}}\right)\geq {\frac {a}{a+b}}f(a+b)+{\frac {b}{a+b}}f(a+b)=f(a+b)}</annotation>

</semantics>

</math></span></span></li></ul></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Functions_of_n_variables">Functions of <i>n</i> variables</h3></div>

<ol><li>A function <span class="texhtml mvar" style="font-style:italic;">f</span> is concave over a convex set <a href="If_and_only_if" title="If and only if">if and only if</a> the function <span class="texhtml mvar" style="font-style:italic;">−f</span> is a <a href="Convex_function" title="Convex function">convex function</a> over the set.</li>

<li>The sum of two concave functions is itself concave and so is the <a href="Pointwise_minimum" class="mw-redirect" title="Pointwise minimum">pointwise minimum</a> of two concave functions, i.e. the set of concave functions on a given domain form a <a href="Semifield" title="Semifield">semifield</a>.</li>

<li>Near a strict <a href="Local_maximum" class="mw-redirect" title="Local maximum">local maximum</a> in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.</li>

<li>Any <a href="Local_maximum" class="mw-redirect" title="Local maximum">local maximum</a> of a concave function is also a <a href="Global_maximum" class="mw-redirect" title="Global maximum">global maximum</a>. A <i>strictly</i> concave function will have at most one global maximum.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>The functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=-x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=-x^{2}}</annotation>

</semantics>

</math></span><img src="./772f4a162f3b6f9325ef5f9404424969480dcd27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.708ex; height:3.176ex;" alt="{\displaystyle f(x)=-x^{2}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\sqrt {x}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)={\sqrt {x}}}</annotation>

</semantics>

</math></span><img src="./200d7544a5ab10c9b757bc078ebed1c1f1102cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.619ex; height:3.009ex;" alt="{\displaystyle g(x)={\sqrt {x}}}" loading="lazy"></span> are concave on their domains, as their second derivatives <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(x)=-2}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f''(x)=-2}</annotation>

</semantics>

</math></span><img src="./22bb726a4b1a7cf994e85b1dba12d8bca4fc2b1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.666ex; height:3.009ex;" alt="{\displaystyle f''(x)=-2}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g''(x)=-{\frac {1}{4x^{3/2}}}}">

<msup>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle g''(x)=-{\frac {1}{4x^{3/2}}}}</annotation>

</semantics>

</math></span><img src="./956f668fcf7c342c73ca8f0338e3d4e2c08e3a6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.066ex; height:4.009ex;" alt="{\textstyle g''(x)=-{\frac {1}{4x^{3/2}}}}" loading="lazy"></span> are always negative.</li>

<li>The <a href="Logarithm" title="Logarithm">logarithm</a> function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\log {x}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\log {x}}</annotation>

</semantics>

</math></span><img src="./9962cbf2ec885e49f036fa1d0a23c2b47473a2ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.205ex; height:2.843ex;" alt="{\displaystyle f(x)=\log {x}}" loading="lazy"></span> is concave on its domain <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,\infty )}</annotation>

</semantics>

</math></span><img src="./da17102e4ed0886686094ee531df040d2e86352a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (0,\infty )}" loading="lazy"></span>, as its derivative <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{x}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}}</annotation>

</semantics>

</math></span><img src="./68f89eaf83a3811c69adb4bf1119bafd661a4c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{x}}}" loading="lazy"></span> is a strictly decreasing function.</li>

<li>Any <a href="Affine_function" class="mw-redirect" title="Affine function">affine function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax+b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=ax+b}</annotation>

</semantics>

</math></span><img src="./75c655df4be41082c4bba924beab2c1dc27d019c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.913ex; height:2.843ex;" alt="{\displaystyle f(x)=ax+b}" loading="lazy"></span> is both concave and convex, but neither strictly-concave nor strictly-convex.</li>

<li>The <a href="Sine" class="mw-redirect" title="Sine">sine</a> function is concave on the interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\pi ]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,\pi ]}</annotation>

</semantics>

</math></span><img src="./3e2a912eda6ef1afe46a81b518fe9da64a832751.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle [0,\pi ]}" loading="lazy"></span>.</li>

<li>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(B)=\log |B|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(B)=\log |B|}</annotation>

</semantics>

</math></span><img src="./606af10cf4863912755b21ccd9ddd71c3209a704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.367ex; height:2.843ex;" alt="{\displaystyle f(B)=\log |B|}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |B|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |B|}</annotation>

</semantics>

</math></span><img src="./09a17e400743ffd99fb026d81d8298a6e6344352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.058ex; height:2.843ex;" alt="{\displaystyle |B|}" loading="lazy"></span> is the <a href="Determinant" title="Determinant">determinant</a> of a <a href="Nonnegative-definite_matrix" class="mw-redirect" title="Nonnegative-definite matrix">nonnegative-definite matrix</a> <i>B</i>, is concave.<sup id="cite_ref-Cover_1988_6-0" class="reference"><a href="#cite_note-Cover_1988-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<ul><li>Rays bending in the <a href="Computation_of_radiowave_attenuation_in_the_atmosphere" class="mw-redirect" title="Computation of radiowave attenuation in the atmosphere">computation of radiowave attenuation in the atmosphere</a> involve concave functions.</li>

<li>In <a href="Expected_utility" class="mw-redirect" title="Expected utility">expected utility</a> theory for <a href="Choice_under_uncertainty" class="mw-redirect" title="Choice under uncertainty">choice under uncertainty</a>, <a href="Cardinal_utility" title="Cardinal utility">cardinal utility</a> functions of <a href="Risk_aversion" title="Risk aversion">risk averse</a> decision makers are concave.</li>

<li>In <a href="Microeconomic_theory" class="mw-redirect" title="Microeconomic theory">microeconomic theory</a>, <a href="Production_function" title="Production function">production functions</a> are usually assumed to be concave over some or all of their domains, resulting in <a href="Diminishing_returns" title="Diminishing returns">diminishing returns</a> to input factors.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>In <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a> and <a href="Information_theory" title="Information theory">information theory</a>, <a href="Entropy_(information_theory)" title="Entropy (information theory)">entropy</a> is a concave function. In the case of thermodynamic entropy, without phase transition, entropy as a function of extensive variables is strictly concave. If the system can undergo phase transition, and if it is allowed to split into two subsystems of different phase (<a href="Phase_separation" title="Phase separation">phase separation</a>, e.g. boiling), the entropy-maximal parameters of the subsystems will result in a combined entropy precisely on the straight line between the two phases. This means that the "effective entropy" of a system with phase transition is the <a href="Convex_envelope" class="mw-redirect" title="Convex envelope">convex envelope</a> of entropy without phase separation; therefore, the entropy of a system including phase separation will be non-strictly concave.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul>

<p><br>

</p>

<ul><li><a href="Concave_polygon" title="Concave polygon">Concave polygon</a></li>

<li><a href="Jensen's_inequality" title="Jensen's inequality">Jensen's inequality</a></li>

<li><a href="Logarithmically_concave_function" title="Logarithmically concave function">Logarithmically concave function</a></li>

<li><a href="Quasiconcave_function" class="mw-redirect" title="Quasiconcave function">Quasiconcave function</a></li>

<li><a href="Concavification" title="Concavification">Concavification</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFLenhartWorkman2007" class="citation book cs1">Lenhart, S.; Workman, J. T. (2007). <i>Optimal Control Applied to Biological Models</i>. Mathematical and Computational Biology Series. Chapman & Hall/ CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-58488-640-2</bdi>.</cite></span>

</li>

<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFVarian,_Hal_R.1992" class="citation book cs1">Varian, Hal R. (1992). <i>Microeconomic analysis</i> (3rd ed.). New York: Norton. p. 489. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-95735-7</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/24847759">24847759</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRudin1976" class="citation book cs1">Rudin, Walter (1976). <i>Analysis</i>. p. 101.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFGradshteynRyzhikHays1976" class="citation journal cs1">Gradshteyn, I. S.; Ryzhik, I. M.; Hays, D. F. (1976-07-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.3452897">"Table of Integrals, Series, and Products"</a>. <i>Journal of Lubrication Technology</i>. <b>98</b> (3): 479. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.3452897">10.1115/1.3452897</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2305">0022-2305</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFHass,_Joel2017" class="citation book cs1">Hass, Joel (13 March 2017). <i>Thomas' calculus</i>. Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006. (Fourteenth ed.). [United States]. p. 203. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-443898-6</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/965446428">965446428</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)</span></span>

</li>

<li id="cite_note-Cover_1988-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cover_1988_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCoverThomas1988" class="citation journal cs1"><a href="Thomas_M._Cover" title="Thomas M. Cover">Cover, Thomas M.</a>; Thomas, J. A. (1988). "Determinant inequalities via information theory". <i><a href="SIAM_Journal_on_Matrix_Analysis_and_Applications" title="SIAM Journal on Matrix Analysis and Applications">SIAM Journal on Matrix Analysis and Applications</a></i>. <b>9</b> (3): <span class="nowrap">384–</span>392. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0609033">10.1137/0609033</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5491763">5491763</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFPembertonRau2015" class="citation book cs1">Pemberton, Malcolm; Rau, Nicholas (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9j5_DQAAQBAJ&pg=PA363"><i>Mathematics for Economists: An Introductory Textbook</i></a>. Oxford University Press. pp. <span class="nowrap">363–</span>364. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-78499-148-7</bdi>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFCallenCallen1985" class="citation book cs1">Callen, Herbert B.; Callen, Herbert B. (1985). "8.1: Intrinsic Stability of Thermodynamic Systems". <i>Thermodynamics and an introduction to thermostatistics</i> (2nd ed.). New York: Wiley. pp. <span class="nowrap">203–</span>206. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-86256-7</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_References">Further References</h2></div>

<ul><li><cite id="CITEREFCrouzeix2008" class="citation book cs1">Crouzeix, J.-P. (2008). <a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008">"Quasi-concavity"</a>. In Durlauf, Steven N.; Blume, Lawrence E (eds.). <i>The New Palgrave Dictionary of Economics</i> (Second ed.). Palgrave Macmillan. pp. <span class="nowrap">815–</span>816. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1057%2F9780230226203.1375">10.1057/9780230226203.1375</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-333-78676-5</bdi>.</cite></li>

<li><cite id="CITEREFRao2009" class="citation book cs1">Rao, Singiresu S. (2009). <i>Engineering Optimization: Theory and Practice</i>. John Wiley and Sons. p. 779. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-18352-6</bdi>.</cite></li></ul>

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</style><div id="Calculus249" style="font-size:114%;margin:0 4em"><a href="Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous function</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Finite_difference" title="Finite difference">Finite difference</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li>

<li><a href="Graph_of_a_function" title="Graph of a function">Graph of a function</a></li>

<li><a href="Linear_function" title="Linear function">Linear function</a></li>

<li><a href="Radian" title="Radian">Radian</a></li>

<li><a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a></li>

<li><a href="Secant_line" title="Secant line">Secant</a></li>

<li><a href="Slope" title="Slope">Slope</a></li>

<li><a href="Tangent" title="Tangent">Tangent</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li>

<li><a href="Limit_of_a_function" title="Limit of a function">Limit of a function</a>

<ul><li><a href="One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li>

<li><a href="Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li>

<li><a href="Order_of_approximation" title="Order of approximation">Order of approximation</a></li>

<li><a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Derivative" title="Derivative">Derivative</a></li>

<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li>

<li><a href="Differential_operator" title="Differential operator">Differential operator</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>

<li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation</a>

<ul><li><a href="Leibniz's_notation" title="Leibniz's notation">Leibniz's notation</a></li>

<li><a href="Newton's_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li>

<li><a href="Differentiation_rules" title="Differentiation rules">Rules of differentiation</a>

<ul><li><a href="Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li>

<li><a href="Power_rule" title="Power rule">Power</a></li>

<li><a href="Chain_rule" title="Chain rule">Chain</a></li>

<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's</a></li>

<li><a href="Product_rule" title="Product rule">Product</a>

<ul><li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li>

<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li>

<li>Other techniques

<ul><li><a href="Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li>

<li><a href="Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li>

<li><a href="Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li></ul></li>

<li><a href="Stationary_point" title="Stationary point">Stationary points</a>

<ul><li><a href="First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li>

<li><a href="Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li>

<li><a href="Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li>

<li><a href="Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li>

<li>Further applications

<ul><li><a href="Newton's_method" title="Newton's method">Newton's method</a></li>

<li><a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li>

<li><a href="Differential_equation" title="Differential equation">Differential equation</a>

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>

<li><a href="Arc_length" title="Arc length">Arc length</a></li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Integral#Properties" title="Integral">Basic properties</a></li>

<li><a href="Constant_of_integration" title="Constant of integration">Constant of integration</a></li>

<li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a>

<ul><li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li>

<li><a href="Integration_by_parts" title="Integration by parts">Integration by parts</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Integration by substitution</a>

<ul><li><a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li>

<li><a href="Euler_substitution" title="Euler substitution">Euler</a></li>

<li><a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a>

<ul><li><a href="Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li>

<li><a href="Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li>

<li>Volumes

<ul><li><a href="Disc_integration" title="Disc integration">Washer method</a></li>

<li><a href="Shell_integration" title="Shell integration">Shell method</a></li></ul></li>

<li><a href="Integral_equation" title="Integral equation">Integral equation</a></li>

<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Derivatives

<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>

<li><a href="Divergence" title="Divergence">Divergence</a></li>

<li><a href="Gradient" title="Gradient">Gradient</a></li>

<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li>

<li>Basic theorems

<ul><li><a href="Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li>

<li><a href="Green's_theorem" title="Green's theorem">Green's</a></li>

<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>

<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li>Advanced topics

<ul><li><a href="Differential_form" title="Differential form">Differential forms</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Generalized_Stokes'_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li>

<li>Types of series

<ul><li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier</a></li>

<li><a href="Geometric_series" title="Geometric series">Geometric</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>

<li><a href="Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li>

<li><a href="Power_series" title="Power series">Power</a>

<ul><li><a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li>

<li>Tests of convergence

<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>

<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>

<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>

<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet's</a></li>

<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>

<li><a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>

<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br>and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li>

<li><a href="E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li>

<li><a href="Exponential_function" title="Exponential function">Exponential function</a></li>

<li><a href="Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li>

<li><a href="Stirling's_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adequality" title="Adequality">Adequality</a></li>

<li><a href="Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li>

<li><a href="Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li>

<li><a href="Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li>

<li><a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li>

<li><a href="Infinitesimal" title="Infinitesimal">Infinitesimal</a></li>

<li><a href="Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li>

<li><a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>

<li><a href="Fluxion" title="Fluxion">Fluxion</a></li>

<li><a href="Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><i><a href="Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li>

<li><i><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li>

<li><a href="List_of_integrals_of_irrational_algebraic_functions" title="List of integrals of irrational algebraic functions">irrational algebraic functions</a></li>

<li><a href="List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li>

<li><a href="List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li>

<li><a href="List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a>

<ul><li><a href="List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li>

<li><a href="List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a>

<ul><li><a href="List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li>

<li><a href="Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li>

<li><a href="Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_limits" title="List of limits">List of limits</a></li>

<li><a href="Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Complex calculus

<ul><li><a href="Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li>

<li>Differential geometry

<ul><li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Curvature" title="Curvature">Curvature</a></li>

<li><a href="Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li>

<li><a href="Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li></ul></li>

<li><a href="Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li>

<li><a href="Gabriel's_horn" title="Gabriel's horn">Gabriel's horn</a></li>

<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>

<li><a href="Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li>

<li><a href="Regiomontanus'_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li>

<li><a href="Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Convex_analysis_and_variational_analysis174" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_analysis_and_variational_analysis174" style="font-size:114%;margin:0 4em"><a href="Convex_analysis" title="Convex analysis">Convex analysis</a> and <a href="Variational_analysis" title="Variational analysis">variational analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_combination" title="Convex combination">Convex combination</a></li>

<li><a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Convex_set" title="Convex set">Convex set</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_convexity_topics" title="List of convexity topics">Topics (list)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Convex_geometry" title="Convex geometry">Convex geometry</a></li>

<li><a href="Convex_metric_space" title="Convex metric space">Convex metric space</a></li>

<li><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></li>

<li><a href="Duality_(optimization)" title="Duality (optimization)">Duality</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a></li>

<li><a href="Simplex" title="Simplex">Simplex</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_conjugate" title="Convex conjugate">Convex conjugate</a></li>

<li>(<a href="Closed_convex_function" title="Closed convex function">Closed</a></li>

<li><a href="Logarithmically_convex_function" title="Logarithmically convex function">Logarithmically</a></li>

<li><a href="Proper_convex_function" title="Proper convex function">Proper</a></li>

<li><a href="Pseudoconvex_function" title="Pseudoconvex function">Pseudo-</a></li>

<li><a href="Quasiconvex_function" title="Quasiconvex function">Quasi-</a>) <a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Invex_function" title="Invex function">Invex function</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Semi-continuity" title="Semi-continuity">Semi-continuity</a></li>

<li><a href="Subderivative" title="Subderivative">Subderivative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results (list)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carath%C3%A9odory's_theorem_(convex_hull)" title="Carathéodory's theorem (convex hull)">Carathéodory's theorem</a></li>

<li><a href="Ekeland's_variational_principle" title="Ekeland's variational principle">Ekeland's variational principle</a></li>

<li><a href="Fenchel%E2%80%93Moreau_theorem" title="Fenchel–Moreau theorem">Fenchel–Moreau theorem</a></li>

<li><a href="Fenchel-Young_inequality" class="mw-redirect" title="Fenchel-Young inequality">Fenchel-Young inequality</a></li>

<li><a href="Jensen's_inequality" title="Jensen's inequality">Jensen's inequality</a></li>

<li><a href="Hermite%E2%80%93Hadamard_inequality" title="Hermite–Hadamard inequality">Hermite–Hadamard inequality</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman theorem</a></li>

<li><a href="Mazur's_lemma" title="Mazur's lemma">Mazur's lemma</a></li>

<li><a href="Shapley%E2%80%93Folkman_lemma" title="Shapley–Folkman lemma">Shapley–Folkman lemma</a></li>

<li><a href="Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson–Ursescu</a></li>

<li><a href="Ursescu_theorem#Simons'_theorem" title="Ursescu theorem">Simons</a></li>

<li><a href="Ursescu_theorem" title="Ursescu theorem">Ursescu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_hull" title="Convex hull">Convex hull</a></li>

<li>(<a href="Orthogonally_convex_set" class="mw-redirect" title="Orthogonally convex set">Orthogonally</a>, <a href="Pseudoconvexity" title="Pseudoconvexity">Pseudo-</a>) <a href="Convex_set" title="Convex set">Convex set</a></li>

<li><a href="Effective_domain" title="Effective domain">Effective domain</a></li>

<li><a href="Epigraph_(mathematics)" title="Epigraph (mathematics)">Epigraph</a></li>

<li><a href="Hypograph_(mathematics)" title="Hypograph (mathematics)">Hypograph</a></li>

<li><a href="John_ellipsoid" title="John ellipsoid">John ellipsoid</a></li>

<li><a href="Radial_set" title="Radial set">Radial set</a>/<a href="Algebraic_interior" title="Algebraic interior">Algebraic interior</a></li>

<li><a href="Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> (<a href="Convex_series#Types_of_subsets" title="Convex series">(cs, lcs)-closed</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(cs, bcs)-complete</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(lower) ideally convex</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(H<i>x</i>)</a>, and <a href="Convex_series#Types_of_subsets" title="Convex series">(Hw<i>x</i>)</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Duality_(optimization)" title="Duality (optimization)">Duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dual_system" title="Dual system">Dual system</a></li>

<li><a href="Duality_gap" title="Duality gap">Duality gap</a></li>

<li><a href="Strong_duality" title="Strong duality">Strong duality</a></li>

<li><a href="Weak_duality" title="Weak duality">Weak duality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications and related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convexity_in_economics" title="Convexity in economics">Convexity in economics</a></li></ul>

</div></td></tr></tbody></table></div>

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