Category: Convex and Concave Functions
Definition of Convex and Concave Functions
Consider two functions on the figure to the right.
They are both increasing, but their form is different.
That's because one of them is convex and another is concave.
Definition. Function that is defined and continuous on interval is called convex (or convex downward or concave upward) if for any and from and numbers and such that and we have that .
Properties of Convex Functions
Here we will talk about properties of convex (or concave upward) function.
We already noted that if function is concave upward then is concave downward. So, these properties also hold for concave downward functions.
Conditions of Concavity (Convexity) of the Function
Often it is very hard to prove convexity (or concavity) of function through definition.
We need more powerful methods.
Fact 1. Suppose that function is defined and continuous on interval , and has finite derivative inside it. Function is concave upward (downward) on if and only if derivative is non-decreasing (non-increasing). Function is strictly concave upward (downward) on if and only if derivative is increasing (decreasing).
Inflection Points
Definition. Point is an inflection point of function if function at this point changes direction of concavity (i.e. from concave upward becomes concave downward or from concave downward becomes concave upward).