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A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. [3] [4] Points where concavity changes (between concave and convex) are inflection points. [5]
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [3] [4] [5] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph .
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Specifically, a twice-differentiable function f is concave up if ″ > and concave down if ″ <. Note that if f ( x ) = x 4 {\displaystyle f(x)=x^{4}} , then x = 0 {\displaystyle x=0} has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is ...
If is concave down in the interval between and , the approximation will be an overestimate (since the derivative is decreasing in that interval). If is concave up, the approximation will be an underestimate. [1]
Define two moves: a left move and a right move, valid on the unit interval as = and () = + and = + and () = The question mark function then obeys a left-move symmetry ? =? and a right-move symmetry ? =? where denotes function composition. These can be arbitrarily concatenated.
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x 2 /2 is a concave function of x. But f is not concave since the second derivative is positive for | x | > 1:
If f″(x) < 0, the stationary point at x is concave down; a maximal extremum. If f″(x) > 0, the stationary point at x is concave up; a minimal extremum. If f″(x) = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point.