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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, a `!concave function`! 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 `F33f`_`[hypograph`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hypograph_(mathematics)]`_`f is convex. The class of concave functions is in a sense the opposite of the class of `F33f`_`[convex functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_function]`_`f. A concave function is also `F33f`_`[synonymously`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Synonym]`_`f called `!concave downwards`!, `!concave down`!, `!convex upwards`!, `!convex cap`!, or `!upper convex`!.

>>Contents

• `F0af`_`[Definition`#definition]`_`f

• `F0af`_`[Properties`#properties]`_`f

• `F0af`_`[Functions of a single variable`#functions-of-a-single-variable]`_`f

• `F0af`_`[Functions of n variables`#functions-of-n-variables]`_`f

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Applications`#applications]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[Further References`#further-references]`_`f

-─

>>Definition

A real-valued `F33f`_`[function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f f {\\displaystyle f} on an `F33f`_`[interval`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Interval_(mathematics)]`_`f (or, more generally, a `F33f`_`[convex set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_set]`_`f in `F33f`_`[vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_space]`_`f) is said to be `*concave`* if, for any x {\\displaystyle x} and y {\\displaystyle y} in the interval and for any α α ∈ ∈ [ 0 , 1 ] {\\displaystyle \\alpha \\in [0,1]} ,`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]

f ( ( 1 − − α α ) x + α α y ) ≥ ≥ ( 1 − − α α ) f ( x ) + α α f ( y ) {\\displaystyle f((1-\\alpha )x+\\alpha y)\\geq (1-\\alpha )f(x)+\\alpha f(y)}

A function is called `*strictly concave`* if

f ( ( 1 − − α α ) x + α α y ) > ( 1 − − α α ) f ( x ) + α α f ( y ) {\\displaystyle f((1-\\alpha )x+\\alpha y)>(1-\\alpha )f(x)+\\alpha f(y)}

for any α α ∈ ∈ ( 0 , 1 ) {\\displaystyle \\alpha \\in (0,1)} and x ≠ ≠ y {\\displaystyle x\\neq y} .

For a function f : R → → R {\\displaystyle f:\\mathbb {R} \\to \\mathbb {R} } , this second definition merely states that for every z {\\displaystyle z} strictly between x {\\displaystyle x} and y {\\displaystyle y} , the point ( z , f ( z ) ) {\\displaystyle (z,f(z))} on the graph of f {\\displaystyle f} is above the straight line joining the points ( x , f ( x ) ) {\\displaystyle (x,f(x))} and ( y , f ( y ) ) {\\displaystyle (y,f(y))} .

A function f {\\displaystyle f} is `F33f`_`[quasiconcave`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quasiconvex_function]`_`f if the upper contour sets of the function S ( a ) = { x : f ( x ) ≥ ≥ a } {\\displaystyle S(a)=\\{x:f(x)\\geq a\\}} are convex sets.`:cite-ref-0-2-0[`F5bf`_`[2`#cite-note-0-2]`_`f]

>>Properties

>>>Functions of a single variable

1. A `F33f`_`[differentiable function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differentiable_function]`_`f f is (strictly) concave on an `F33f`_`[interval`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Interval_(mathematics)]`_`f if and only if its `F33f`_`[derivative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Derivative]`_`f function f ′ is (strictly) `F33f`_`[monotonically decreasing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monotonically_decreasing]`_`f on that interval, that is, a concave function has a non-increasing (decreasing) `F33f`_`[slope`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Slope]`_`f.`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]

2. `F33f`_`[Points`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Point_(geometry)]`_`f where concavity changes (between concave and `F33f`_`[convex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_function]`_`f) are `F33f`_`[inflection points`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inflection_point]`_`f.`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f]

3. If f is twice-`F33f`_`[differentiable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differentiable_function]`_`f, then f is concave `F33f`_`[if and only if`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=If_and_only_if]`_`f f ′′ is `F33f`_`[non-positive`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Non-positive]`_`f (or, informally, if the "`F33f`_`[acceleration`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Acceleration]`_`f" is non-positive). If f ′′ is `F33f`_`[negative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Negative_numbers]`_`f then f is strictly concave, but the converse is not true, as shown by `*f`*(`*x`*) = −`*x`*4.

4. If f is concave and differentiable, then it is bounded above by its first-order `F33f`_`[Taylor approximation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taylor_approximation]`_`f:`:cite-ref-0-2-1[`F5bf`_`[2`#cite-note-0-2]`_`f] f ( y ) ≤ ≤ f ( x ) + f ′ ( x ) [ y − − x ] {\\displaystyle f(y)\\leq f(x)+f'(x)[y-x]}

5. A `F33f`_`[Lebesgue measurable function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lebesgue_measurable_function]`_`f on an interval `!C`! is concave `F33f`_`[if and only if`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=If_and_only_if]`_`f it is midpoint concave, that is, for any x and y in `!C`! f ( x + y 2 ) ≥ ≥ f ( x ) + f ( y ) 2 {\\displaystyle f\\left({\\frac {x+y}{2}}\\right)\\geq {\\frac {f(x)+f(y)}{2}}}

6. If a function f is concave, and `*f`*(0) ≥ 0, then f is `F33f`_`[subadditive`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Subadditivity]`_`f on [ 0 , ∞ ∞ ) {\\displaystyle [0,\\infty )} . Proof:

• Since f is concave and 1 ≥ t ≥ 0, letting `*y`* = 0 we have f ( t x ) = f ( t x + ( 1 − − t ) ⋅ ⋅ 0 ) ≥ ≥ t f ( x ) + ( 1 − − t ) f ( 0 ) ≥ ≥ t f ( x ) . {\\displaystyle f(tx)=f(tx+(1-t)\\cdot 0)\\geq tf(x)+(1-t)f(0)\\geq tf(x).}

• For a , b ∈ ∈ [ 0 , ∞ ∞ ) {\\displaystyle a,b\\in [0,\\infty )} : f ( a ) + f ( b ) = f ( ( a + b ) a a + b ) + f ( ( a + b ) b a + b ) ≥ ≥ a a + b f ( a + b ) + b a + b f ( a + b ) = f ( a + b ) {\\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)}

>>>Functions of n variables

1. A function f is concave over a convex set `F33f`_`[if and only if`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=If_and_only_if]`_`f the function −f is a `F33f`_`[convex function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_function]`_`f over the set.

2. The sum of two concave functions is itself concave and so is the `F33f`_`[pointwise minimum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pointwise_minimum]`_`f of two concave functions, i.e. the set of concave functions on a given domain form a `F33f`_`[semifield`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Semifield]`_`f.

3. Near a strict `F33f`_`[local maximum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_maximum]`_`f 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.

4. Any `F33f`_`[local maximum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_maximum]`_`f of a concave function is also a `F33f`_`[global maximum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Global_maximum]`_`f. A `*strictly`* concave function will have at most one global maximum.

>>Examples

• The functions f ( x ) = − − x 2 {\\displaystyle f(x)=-x^{2}} and g ( x ) = x {\\displaystyle g(x)={\\sqrt {x}}} are concave on their domains, as their second derivatives f ″ ( x ) = − − 2 {\\displaystyle f''(x)=-2} and g ″ ( x ) = − − 1 4 x 3 / 2 {\\textstyle g''(x)=-{\\frac {1}{4x^{3/2}}}} are always negative.

• The `F33f`_`[logarithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithm]`_`f function f ( x ) = log ⁡ ⁡ x {\\displaystyle f(x)=\\log {x}} is concave on its domain ( 0 , ∞ ∞ ) {\\displaystyle (0,\\infty )} , as its derivative 1 x {\\displaystyle {\\frac {1}{x}}} is a strictly decreasing function.

• Any `F33f`_`[affine function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Affine_function]`_`f f ( x ) = a x + b {\\displaystyle f(x)=ax+b} is both concave and convex, but neither strictly-concave nor strictly-convex.

• The `F33f`_`[sine`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sine]`_`f function is concave on the interval [ 0 , π π ] {\\displaystyle [0,\\pi ]} .

• The function f ( B ) = log ⁡ ⁡ | B | {\\displaystyle f(B)=\\log |B|} , where | B | {\\displaystyle |B|} is the `F33f`_`[determinant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Determinant]`_`f of a `F33f`_`[nonnegative-definite matrix`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonnegative-definite_matrix]`_`f `*B`*, is concave.`:cite-ref-cover-1988-6-0[`F5bf`_`[6`#cite-note-cover-1988-6]`_`f]

>>Applications

• Rays bending in the `F33f`_`[computation of radiowave attenuation in the atmosphere`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Computation_of_radiowave_attenuation_in_the_atmosphere]`_`f involve concave functions.

• In `F33f`_`[expected utility`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Expected_utility]`_`f theory for `F33f`_`[choice under uncertainty`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Choice_under_uncertainty]`_`f, `F33f`_`[cardinal utility`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cardinal_utility]`_`f functions of `F33f`_`[risk averse`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Risk_aversion]`_`f decision makers are concave.

• In `F33f`_`[microeconomic theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Microeconomic_theory]`_`f, `F33f`_`[production functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Production_function]`_`f are usually assumed to be concave over some or all of their domains, resulting in `F33f`_`[diminishing returns`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Diminishing_returns]`_`f to input factors.`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f]

• In `F33f`_`[thermodynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thermodynamics]`_`f and `F33f`_`[information theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Information_theory]`_`f, `F33f`_`[entropy`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Entropy_(information_theory)]`_`f 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 (`F33f`_`[phase separation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Phase_separation]`_`f, 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 `F33f`_`[convex envelope`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_envelope]`_`f of entropy without phase separation; therefore, the entropy of a system including phase separation will be non-strictly concave.`:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f]

>>See also

• `F33f`_`[Concave polygon`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Concave_polygon]`_`f

• `F33f`_`[Jensen's inequality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Jensen's_inequality]`_`f

• `F33f`_`[Logarithmically concave function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmically_concave_function]`_`f

• `F33f`_`[Quasiconcave function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quasiconcave_function]`_`f

• `F33f`_`[Concavification`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Concavification]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citereflenhartworkman2007`aLenhart, S.; Workman, J. T. (2007). `*Optimal Control Applied to Biological Models`*. Mathematical and Computational Biology Series. Chapman & Hall/ CRC. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-58488-640-2.

`:cite-note-0-2`!2.`! `F0af`_`[↑`#cite-ref-0-2-0]`_`f `:citerefvarian-hal-r-1992`aVarian, Hal R. (1992). `*Microeconomic analysis`* (3rd ed.). New York: Norton. p. 489. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-393-95735-7. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 24847759.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefrudin1976`aRudin, Walter (1976). `*Analysis`*. p. 101.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citerefgradshteynryzhikhays1976`aGradshteyn, I. S.; Ryzhik, I. M.; Hays, D. F. (1976-07-01). "Table of Integrals, Series, and Products". `*Journal of Lubrication Technology`*. `!98`! (3): 479. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1115/1.3452897. `F33f`_`[ISSN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISSN_(identifier)]`_`f 0022-2305.

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f `:citerefhass-joel2017`aHass, Joel (13 March 2017). `*Thomas' calculus`*. Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006. (Fourteenth ed.). [United States]. p. 203. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-13-443898-6. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 965446428.`B100`F9d9{{cite book}}`f`b: CS1 maint: location missing publisher (link)

`:cite-note-cover-1988-6`!6.`! `F0af`_`[↑`#cite-ref-cover-1988-6-0]`_`f `:citerefcoverthomas1988`a`F33f`_`[Cover, Thomas M.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thomas_M._Cover]`_`f; Thomas, J. A. (1988). "Determinant inequalities via information theory". `*`F33f`_`[SIAM Journal on Matrix Analysis and Applications`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=SIAM_Journal_on_Matrix_Analysis_and_Applications]`_`f`*. `!9`! (3): 384–392. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1137/0609033. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 5491763.

`:cite-note-7`!7.`! `F0af`_`[↑`#cite-ref-7]`_`f `:citerefpembertonrau2015`aPemberton, Malcolm; Rau, Nicholas (2015). `*Mathematics for Economists: An Introductory Textbook`*. Oxford University Press. pp. 363–364. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-78499-148-7.

`:cite-note-8`!8.`! `F0af`_`[↑`#cite-ref-8]`_`f `:citerefcallencallen1985`aCallen, Herbert B.; Callen, Herbert B. (1985). "8.1: Intrinsic Stability of Thermodynamic Systems". `*Thermodynamics and an introduction to thermostatistics`* (2nd ed.). New York: Wiley. pp. 203–206. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-471-86256-7.

>>Further References

• `:citerefcrouzeix2008`aCrouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E (eds.). `*The New Palgrave Dictionary of Economics`* (Second ed.). Palgrave Macmillan. pp. 815–816. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1057/9780230226203.1375. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-333-78676-5.

• `:citerefrao2009`aRao, Singiresu S. (2009). `*Engineering Optimization: Theory and Practice`*. John Wiley and Sons. p. 779. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-470-18352-6.

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