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<title>Maximum and minimum</title>

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Maximum and minimum

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</style><div role="note" class="hatnote navigation-not-searchable">"Extreme value" redirects here. For other uses, see <a href="Extreme_value_(disambiguation)" class="mw-disambig" title="Extreme value (disambiguation)">Extreme value (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Maximum" and "Minimum" redirect here. For other uses, see <a href="Maximum_(disambiguation)" class="mw-disambig" title="Maximum (disambiguation)">Maximum (disambiguation)</a> and <a href="Minimum_(disambiguation)" class="mw-disambig" title="Minimum (disambiguation)">Minimum (disambiguation)</a>.</div>

In <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the maximum and minimum<a href="#cite_note-1">[a]</a> of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> are, respectively, the greatest and least value taken by the function. Known generically as extremum,<a href="#cite_note-2">[b]</a> they may be defined either within a given <a href="Interval_(mathematics)" title="Interval (mathematics)">range</a> (the local or relative extrema) or on the entire <a href="Domain_of_a_function" title="Domain of a function">domain</a> (the global or absolute extrema) of a function.<a href="#cite_note-3">[1]</a><a href="#cite_note-4">[2]</a><a href="#cite_note-5">[3]</a> <a href="Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> was one of the first mathematicians to propose a general technique, <a href="Adequality" title="Adequality">adequality</a>, for finding the maxima and minima of functions.

As defined in <a href="Set_theory" title="Set theory">set theory</a>, the maximum and minimum of a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> are the <a href="Greatest_and_least_elements" class="mw-redirect" title="Greatest and least elements">greatest and least elements</a> in the set, respectively. Unbounded <a href="Infinite_set" title="Infinite set">infinite sets</a>, such as the set of <a href="Real_number" title="Real number">real numbers</a>, have no minimum or maximum.

In <a href="Statistics" title="Statistics">statistics</a>, the corresponding concept is the <a href="Sample_maximum_and_minimum" title="Sample maximum and minimum">sample maximum and minimum</a>.

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

A real-valued <a href="Function_(mathematics)" title="Function (mathematics)">function</a> f defined on a <a href="Domain_of_a_function" title="Domain of a function">domain</a> X has a global (or absolute) maximum point at x∗, if f(x∗) ≥ f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x∗, if f(x∗) ≤ f(x) for all x in X. The value of the function at a maximum point is called the <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>maximum value of the function, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(f(x))}">

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

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</math><img src="./2c1e9b1277019f9867f6c9cb856485f3f430ba0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.553ex; height:2.843ex;" alt="{\displaystyle \max(f(x))}" loading="lazy">, and the value of the function at a minimum point is called the minimum value of the function, (denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min(f(x))}">

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

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</math><img src="./04384f84f55ba4e5a9d87b3948d30c9ed06200ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.102ex; height:2.843ex;" alt="{\displaystyle \min(f(x))}" loading="lazy"> for clarity). Symbolically, this can be written as follows:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in X}</annotation>

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</math><img src="./1b79e955b57dd7aada93b8afd459996ae941d480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}" loading="lazy"> is a global maximum point of function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {R} ,}">

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

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</math><img src="./e72b93833f90a2d703ef1c01556e957bb187f351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.135ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} ,}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}">

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<annotation encoding="application/x-tex">{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}</annotation>

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</math><img src="./c2272de3591ea6428f4e6d0620cffb5e63cba762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.274ex; height:2.843ex;" alt="{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}" loading="lazy"></dd></dl>

The definition of global minimum point also proceeds similarly.

If the domain X is a <a href="Metric_space" title="Metric space">metric space</a>, then f is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) for all x in X within distance ε of x∗. Similarly, the function has a local minimum point at x∗, if f(x∗) ≤ f(x) for all x in X within distance ε of x∗. A similar definition can be used when X is a <a href="Topological_space" title="Topological space">topological space</a>, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:

<msub>

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

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</math><img src="./c9b63a9cf5929b2fe7fca007865e043bb9004c21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.664ex; height:2.843ex;" alt="{\displaystyle (X,d_{X})}" loading="lazy"> be a metric space and function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {R} }">

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

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</math><img src="./669fa4832da4b0b229d77eadb270e95188f2eb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} }" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in X}</annotation>

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</math><img src="./1b79e955b57dd7aada93b8afd459996ae941d480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}" loading="lazy"> is a local maximum point of function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><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"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists \varepsilon >0)}">

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

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</math><img src="./1f09b0440cc9530512b4972ab5f3aa697e3616c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.446ex; height:2.843ex;" alt="{\displaystyle (\exists \varepsilon >0)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}</annotation>

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</math><img src="./ff6ea2c6488ad08285bb2a20c508de7f2f32a94d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.24ex; height:2.843ex;" alt="{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}" loading="lazy"></dd></dl>

The definition of local minimum point can also proceed similarly.

In both the global and local cases, the concept of a strict extremum can be defined. For example, x∗ is a strict global maximum point if for all x in X with x ≠ x∗, we have f(x∗) > f(x), and x∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x∗ with x ≠ x∗, we have f(x∗) > f(x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.

A <a href="Continuous_function" title="Continuous function">continuous</a> real-valued function with a <a href="Compact_space" title="Compact space">compact</a> domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> of <a href="Real_number" title="Real number">real numbers</a> (see the graph above).

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

Finding global maxima and minima is the goal of <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>. If a function is continuous on a closed interval, then by the <a href="Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a>, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one.

For <a href="Differentiable_functions" class="mw-redirect" title="Differentiable functions">differentiable functions</a>, <a href="Fermat's_theorem_(stationary_points)" class="mw-redirect" title="Fermat's theorem (stationary points)">Fermat's theorem</a> states that local extrema in the interior of a domain must occur at <a href="Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> (or points where the derivative equals zero).<a href="#cite_note-6">[4]</a> However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the <a href="First_derivative_test" class="mw-redirect" title="First derivative test">first derivative test</a>, <a href="Derivative_test#Second-derivative_test_(single_variable)" title="Derivative test">second derivative test</a>, or <a href="Higher-order_derivative_test" class="mw-redirect" title="Higher-order derivative test">higher-order derivative test</a>, given sufficient differentiability.<a href="#cite_note-7">[5]</a>

For any function that is defined <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a>, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least).

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

<th>Function</th>

<th>Maxima and minima

</th></tr>

<tr>

<td>Unique global minimum at x = 0.

</td></tr>

<tr>

<td>No global minima or maxima. Although the first derivative (3x2) is 0 at x = 0, this is an <a href="Inflection_point" title="Inflection point">inflection point</a>. (2nd derivative is 0 at that point.)

</td></tr>

<tr>

<mroot>

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</mroot>

</mrow>

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

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</math><img src="./61b4a0a76158849854a302fc639dfc882ec16008.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{x}]{x}}}" loading="lazy"></big></td>

<td>Unique global maximum at x = <a href="E_(mathematical_constant)" title="E (mathematical constant)">e</a>. (See figure at right)

</td></tr>

<tr>

<td>Unique global maximum over the positive real numbers at x = 1/e.

</td></tr>

<tr>

<td>First derivative x2 − 1 and <a href="Second_derivative" title="Second derivative">second derivative</a> 2x. Setting the first derivative to 0 and solving for x gives <a href="Stationary_point" title="Stationary point">stationary points</a> at −1 and +1. From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum. This function has no global maximum or minimum.

</td></tr>

<tr>

<td>Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.

</td></tr>

<tr>

<td>Infinitely many global maxima at 0, ±2π, ±4π, ..., and infinitely many global minima at ±π, ±3π, ±5π, ....

</td></tr>

<tr>

<td>Infinitely many local maxima and minima, but no global maximum or minimum.

</td></tr>

<tr>

<td>Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)

</td></tr>

<tr>

<td>x3 + 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2]</td>

<td>Local maximum at x = −1−√15/3, local minimum at x = −1+√15/3, global maximum at x = 2 and global minimum at x = −4.

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

For a practical example,<a href="#cite_note-minimization_maximization_refresher-8">[6]</a> assume a situation where someone has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 200}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 200}</annotation>

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</math><img src="./198350442fbcb39391818f80732eb8701439bad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 200}" loading="lazy"> feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><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"> is the length, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><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"> is the width, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

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

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</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"> is the area:

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<annotation encoding="application/x-tex">{\displaystyle 2x+2y=200}</annotation>

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</math><img src="./441dd355ed8fa0eb140ee74f053bc221f6d9048c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.236ex; height:2.509ex;" alt="{\displaystyle 2x+2y=200}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle 2y=200-2x}</annotation>

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</math><img src="./74952699b03f5961e3303d3b23088316032cb7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.236ex; height:2.509ex;" alt="{\displaystyle 2y=200-2x}" loading="lazy"></dd>

<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}}</annotation>

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</math><img src="./7429d8a609e5bdbe2296931996ae5a0c022d7a33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.909ex; height:5.343ex;" alt="{\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle y=100-x}</annotation>

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</math><img src="./e420c14cad943cd21a14d73b2ee682f2bfb69883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.911ex; height:2.509ex;" alt="{\displaystyle y=100-x}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle xy=x(100-x)}</annotation>

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The derivative with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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<mtr>

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<mtr>

<mtd>

<mfrac>

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<mrow>

<mrow>

<msup>

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</msup>

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}}</annotation>

</semantics>

</math><img src="./927ccfa5c8f0fb40bee42957fd803189e72e8466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.657ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}}" loading="lazy"></dd></dl>

Setting this equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

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

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<annotation encoding="application/x-tex">{\displaystyle 0=100-2x}</annotation>

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</math><img src="./c146b77770d725bdd0923a90792f5310abe944e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.081ex; height:2.343ex;" alt="{\displaystyle 0=100-2x}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle 2x=100}</annotation>

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</math><img src="./928347e70fe9590f6fd49a9c23694b79f622a4b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.078ex; height:2.176ex;" alt="{\displaystyle 2x=100}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle x=50}</annotation>

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</math><img src="./0c97c19892373defe73b8a95b63f3840d574da16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle x=50}" loading="lazy"></dd></dl>

reveals that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=50}">

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<annotation encoding="application/x-tex">{\displaystyle x=50}</annotation>

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Now retrieve the <a href="Interval_(mathematics)" title="Interval (mathematics)">endpoints</a> by determining the interval to which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

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

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</math><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"> is restricted. Since width is positive, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

</mstyle>

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

</semantics>

</math><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy">, and since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=100-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=100-y}</annotation>

</semantics>

</math><img src="./359b5165e59b56b580004ee749c52ea07f68befd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.911ex; height:2.509ex;" alt="{\displaystyle x=100-y}" loading="lazy">, that implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<100}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<100}</annotation>

</semantics>

</math><img src="./c975b633e15b0e7f27986d41e17014bfcc4fb433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.916ex; height:2.176ex;" alt="{\displaystyle x<100}" loading="lazy">.

Plug in critical point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 50}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 50}</annotation>

</semantics>

</math><img src="./17e5f8966bed37734cd86d4fd3c302913bb6d48b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 50}" loading="lazy">, as well as endpoints <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 100}</annotation>

</semantics>

</math><img src="./0572cd017c6d7936a12737c9d614a2f801f94a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}" loading="lazy">, into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=x(100-x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy=x(100-x)}</annotation>

</semantics>

</math><img src="./a54468a2c8da1263d601178d3c34da9808e0dfba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.38ex; height:2.843ex;" alt="{\displaystyle xy=x(100-x)}" loading="lazy">, and the results are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2500,0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2500,0,}</annotation>

</semantics>

</math><img src="./e19ae76eed1f2872da577bd5f24e666f871419e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.493ex; height:2.509ex;" alt="{\displaystyle 2500,0,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

Therefore, the greatest area attainable with a rectangle of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 200}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 200}</annotation>

</semantics>

</math><img src="./198350442fbcb39391818f80732eb8701439bad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 200}" loading="lazy"> feet of fencing is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 50\times 50=2500}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 50\times 50=2500}</annotation>

</semantics>

</math><img src="./ab081230ede8044155c48dec8c781f04209bcbd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.239ex; height:2.176ex;" alt="{\displaystyle 50\times 50=2500}" loading="lazy">.<a href="#cite_note-minimization_maximization_refresher-8">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Functions_of_more_than_one_variable">Functions of more than one variable</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Second_partial_derivative_test" title="Second partial derivative test">Second partial derivative test</a></div>

For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first <a href="Partial_derivatives" class="mw-redirect" title="Partial derivatives">partial derivatives</a> as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a <a href="Saddle_point" title="Saddle point">saddle point</a>. For use of these conditions to solve for a maximum, the function z must also be <a href="Differentiable_function" title="Differentiable function">differentiable</a> throughout. The <a href="Second_partial_derivative_test" title="Second partial derivative test">second partial derivative test</a> can help classify the point as a relative maximum or relative minimum.

In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the <a href="Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a> and <a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a> to prove this by <a href="Proof_by_contradiction" title="Proof by contradiction">contradiction</a>). In two and more dimensions, this argument fails. This is illustrated by the function

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./70b4a21ac7525541b799f233d44a9501ba76bd72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.704ex; height:3.176ex;" alt="{\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,}" loading="lazy"></dd></dl>

whose only critical point is at (0,0), which is a local minimum with f(0,0) = 0. However, it cannot be a global one, because f(2,3) = −5.

<div class="mw-heading mw-heading2"><h2 id="Maxima_or_minima_of_a_functional">Maxima or minima of a functional</h2></div>

If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a <a href="Functional_(mathematics)" title="Functional (mathematics)">functional</a>), then the extremum is found using the <a href="Calculus_of_variations" title="Calculus of variations">calculus of variations</a>.

<div class="mw-heading mw-heading2"><h2 id="In_relation_to_sets">In relation to sets</h2></div>

Maxima and minima can also be defined for sets. In general, if an <a href="Ordered_set" class="mw-redirect" title="Ordered set">ordered set</a> S has a <a href="Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a> m, then m is a <a href="Maximal_element" class="mw-redirect" title="Maximal element">maximal element</a> of the set, also denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(S)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./08446be939ab20ba0b116419017fd087c60af3e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.634ex; height:2.843ex;" alt="{\displaystyle \max(S)}" loading="lazy">. Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T), then m is a <a href="Supremum" class="mw-redirect" title="Supremum">least upper bound</a> of S in T. Similar results hold for <a href="Least_element" class="mw-redirect" title="Least element">least element</a>, <a href="Minimal_element" class="mw-redirect" title="Minimal element">minimal element</a> and <a href="Infimum" class="mw-redirect" title="Infimum">greatest lower bound</a>. The maximum and minimum function for sets are used in <a href="Database" title="Database">databases</a>, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-<a href="Decomposable_aggregation_function" class="mw-redirect" title="Decomposable aggregation function">decomposable aggregation functions</a>.

In the case of a general <a href="Partial_order" class="mw-redirect" title="Partial order">partial order</a>, a least element (i.e., one that is less than all others) should not be confused with the minimal element (nothing is lesser). Likewise, a <a href="Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a> of a <a href="Partially_ordered_set" title="Partially ordered set">partially ordered set</a> (poset) is an <a href="Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> of the set which is contained within the set, whereas the maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A), then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.

In a <a href="Total_order" title="Total order">totally ordered</a> set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms minimum and maximum.

If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have a maximum or a minimum. For example, the set of <a href="Natural_number" title="Natural number">natural numbers</a> has no maximum, though it has a minimum. If an infinite chain S is bounded, then the <a href="Topological_closure" class="mw-redirect" title="Topological closure">closure</a> Cl(S) of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set S, respectively.

<div class="mw-heading mw-heading2"><h2 id="Argument_of_the_maximum">Argument of the maximum</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Arg_max" title="Arg max">Arg max</a>.[<a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Arg_max&action=edit">edit</a>]</div><div class="excerpt">

In <a href="Mathematics" title="Mathematics">mathematics</a>, the arguments of the maxima (abbreviated <a href="Arg_max" title="Arg max">arg max</a> or argmax) and arguments of the minima (abbreviated arg min or argmin) are the input points at which a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> output value is <a href="Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maximized and minimized</a>, respectively.<a href="#cite_note-10">[8]</a> While the <a href="Argument_of_a_function" title="Argument of a function">arguments</a> are defined over the <a href="Domain_of_a_function" title="Domain of a function">domain of a function</a>, the output is part of its <a href="Codomain" title="Codomain">codomain</a>.</div></div>

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

<li><a href="Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a></li>

<li><a href="Limit_superior_and_limit_inferior" class="mw-redirect" title="Limit superior and limit inferior">Limit superior and limit inferior</a></li>

<li><a href="Maximum-minimums_identity" title="Maximum-minimums identity">Maximum-minimums identity</a></li>

<li><a href="Mechanical_equilibrium" title="Mechanical equilibrium">Mechanical equilibrium</a></li>

<li><a href="Mex_(mathematics)" title="Mex (mathematics)">Mex (mathematics)</a></li>

<li><a href="Saddle_point" title="Saddle point">Saddle point</a></li>

<li><a href="Sample_maximum_and_minimum" title="Sample maximum and minimum">Sample maximum and minimum</a></li></ul>

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="Plural" title="Plural">PL</a>: maxima and minima (or maximums and minimums).

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> PL: extrema.

</li>

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

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

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</style><cite id="CITEREFStewart2008" class="citation book cs1"><a href="James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1">Calculus: Early Transcendentals</a> (6th ed.). <a href="Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-495-01166-8</bdi>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFLarsonEdwards2009" class="citation book cs1"><a href="Ron_Larson_(mathematician)" class="mw-redirect" title="Ron Larson (mathematician)">Larson, Ron</a>; Edwards, Bruce H. (2009). Calculus (9th ed.). <a href="Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-547-16702-2</bdi>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFThomasWeirHass2010" class="citation book cs1"><a href="George_B._Thomas" title="George B. Thomas">Thomas, George B.</a>; Weir, Maurice D.; <a href="Joel_Hass" title="Joel Hass">Hass, Joel</a> (2010). Thomas' Calculus: Early Transcendentals (12th ed.). <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-321-58876-0</bdi>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Minimum.html">"Minimum"</a>. mathworld.wolfram.com. Retrieved 2020-08-30.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Maximum.html">"Maximum"</a>. mathworld.wolfram.com. Retrieved 2020-08-30.</cite>

</li>

<li id="cite_note-minimization_maximization_refresher-8">^ <a href="#cite_ref-minimization_maximization_refresher_8-0">a</a> <a href="#cite_ref-minimization_maximization_refresher_8-1">b</a> <cite id="CITEREFGarrett,_Paul" class="citation web cs1">Garrett, Paul. <a rel="nofollow" class="external text" href="https://mathinsight.org/minimization_maximization_refresher">"Minimization and maximization refresher"</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> "<a rel="nofollow" class="external text" href="http://physics.usyd.edu.au/teach_res/mp/doc/math_sinc_function.pdf">The Unnormalized Sinc Function</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170215045226/http://www.physics.usyd.edu.au/teach_res/mp/doc/math_sinc_function.pdf">Archived</a> 2017-02-15 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>", University of Sydney

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> For clarity, we refer to the input (x) as points and the output (y) as values; compare <a href="Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a> and <a href="Critical_value_(critical_point)" class="mw-redirect" title="Critical value (critical point)">critical value</a>.

</li>

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

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Extrema_(calculus)" class="extiw external" title="commons:Category:Extrema (calculus)">Extrema (calculus)</a>.</div></div>

</div>

<div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/maxima" class="extiw external" title="wiktionary:maxima">maxima</a>, <a href="https://en.wiktionary.org/wiki/minima" class="extiw external" title="wiktionary:minima">minima</a>, or <a href="https://en.wiktionary.org/wiki/extremum" class="extiw external" title="wiktionary:extremum">extremum</a> in Wiktionary, the free dictionary.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://www.maa.org/publications/periodicals/convergence/thomas-simpson-and-maxima-and-minima">Thomas Simpson's work on Maxima and Minima</a> at <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070713083148/http://mathdl.maa.org/convergence/1/">Convergence</a></li>

<li><a rel="nofollow" class="external text" href="http://www.mathalino.com/reviewer/differential-calculus/application-of-maxima-and-minima">Application of Maxima and Minima with sub pages of solved problems</a></li>

<li><cite class="citation encyclopaedia cs1">Jolliffe, Arthur Ernest (1911). <a class="external text external" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Maxima_and_Minima">"Maxima and Minima" </a>. <a href="Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a>. Vol. 17 (11th ed.). pp. 918–920.</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="Concave_function" title="Concave function">Concave function</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>

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

<li><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></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><div><div class="zim-footer">

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