Editing Concave function (section)

==Properties==
[[File:cubic_graph_special_points_repeated.svg|thumb|A cubic function is concave (left half) when its first derivative (red) is monotonically decreasing i.e. its second derivative (orange) is negative, and convex (right half) when its first derivative is monotonically increasing i.e. its second derivative is positive]]
===Functions of a single variable===
# A [[differentiable function]] {{mvar|f}} is (strictly) concave on an [[interval (mathematics)|interval]] if and only if its [[derivative]] function {{mvar|f &prime;}} is (strictly) [[monotonically decreasing]] on that interval, that is, a concave function has a non-increasing (decreasing) [[slope]].<ref>{{Cite book| last=Rudin| first=Walter| title=Analysis| year=1976| pages= 101}}</ref><ref>{{Cite journal |last1=Gradshteyn|first1=I. S.| last2=Ryzhik|first2=I. M.| last3=Hays|first3=D. F.| date=1976-07-01| title=Table of Integrals, Series, and Products| journal=Journal of Lubrication Technology| volume=98|issue=3|pages=479| doi=10.1115/1.3452897|issn=0022-2305 |doi-access=free}}</ref>
# [[Point (geometry)|Points]] where concavity changes (between concave and [[convex function|convex]]) are [[inflection point]]s.<ref>{{Cite book|last=Hass, Joel |url=https://www.worldcat.org/oclc/965446428| title=Thomas' calculus| others=Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006.|date=13 March 2017| isbn=978-0-13-443898-6| edition=Fourteenth| location=[United States]| pages=203| oclc=965446428}}</ref>
# If {{mvar|f}} is twice-[[Differentiable function|differentiable]], then {{mvar|f}} is concave [[if and only if]] {{mvar|f &prime;&prime;}} is [[non-positive]] (or, informally, if the "[[acceleration]]" is non-positive). If {{mvar|f &prime;&prime;}} is [[negative numbers|negative]] then {{mvar|f}} is strictly concave, but the converse is not true, as shown by {{math|1=''f''(''x'') = &minus;''x''<sup>4</sup>}}.
# If {{mvar|f}} is concave and differentiable, then it is bounded above by its first-order [[Taylor approximation]]:<ref name=":0" /> <math display="block">f(y) \leq f(x) + f'(x)[y-x]</math>
# A [[Lebesgue measurable function]] on an interval {{math|'''C'''}} is concave [[if and only if]] it is midpoint concave, that is, for any {{mvar|x}} and {{mvar|y}} in {{math|'''C'''}} <math display="block"> f\left( \frac{x+y}2 \right) \ge \frac{f(x) + f(y)}2</math>
# If a function {{mvar|f}} is concave, and {{math|''f''(0) ≥ 0}}, then {{mvar|f}} is [[subadditivity|subadditive]] on <math>[0,\infty)</math>. Proof:
#* Since {{mvar|f}} is concave and {{math|1 ≥ t ≥ 0}}, letting {{math|1=''y'' = 0}} we have <math display="block">f(tx) = f(tx+(1-t)\cdot 0) \ge t f(x)+(1-t)f(0) \ge t f(x) .</math>
#* For <math>a,b\in[0,\infty)</math>: <math display="block">f(a) + f(b) = f \left((a+b) \frac{a}{a+b} \right) + f \left((a+b) \frac{b}{a+b} \right)
\ge \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) = f(a+b)</math>

===Functions of ''n'' variables===
# A function {{mvar|f}} is concave over a convex set [[if and only if]] the function {{mvar|−f}}  is a [[convex function]] over the set.
# The sum of two concave functions is itself concave and so is the [[pointwise minimum]] of two concave functions, i.e. the set of concave functions on a given domain form a [[semifield]].
# Near a strict [[local maximum]] 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.
# Any [[local maximum]] of a concave function is also a [[global maximum]]. A ''strictly'' concave function will have at most one global maximum.