Other Definitions
concavity (dict)
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ConcavityIn geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. Concave functions In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. Similarly, a differentiable function f is concave on an interval if its derivative function f ′ is decreasing on that interval: a concave function has a decreasing slope. A function that is convex is often synonymously called concave upwards, and a function that is concave is often synonymously called concave downward. For a twice-differentiable function f, if the second derivative, f ''(x), is positive (or, if the acceleration is positive), then the graph is convex (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward). Points where concavity changes are inflection points. If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum. Contrary to the impression one may get from a calculus course, differentiability is not essential to these concepts; see convex. In mathematics, a function f(x) is said to be concave on an interval b if, for all x,y in b, -
Additionally, is strictly concave if -
A continuous function on C is concave if and only if -
for any x and y in C. Equivalently, f(x) is concave on b iff the function −f(x) is convex on every subinterval of b. If f(x) is twice-differentiable, then f(x) is concave iff f ′′(x) is non-positive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x4. A function is called quasiconcave iff there is an such that for all |
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