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 $$$y={f{{\left({x}\right)}}}$$$ that is defined and continuous on interval $$$X$$$ is called convex (or convex downward or concave upward) if for any $$$a$$$ and $$$b$$$ from $$$X$$$ and numbers $$${q}_{{1}}$$$ and $$${q}_{{2}}$$$ such that $$${q}_{{1}}+{q}_{{2}}={1}$$$ and $$${q}_{{1}}\ge{0},{q}_{{2}}\ge{0}$$$ we have that $$${f{{\left({q}_{{1}}{a}+{q}_{{2}}{b}\right)}}}\le{q}_{{1}}{f{{\left({a}\right)}}}+{q}_{{2}}{f{{\left({b}\right)}}}$$$.

Properties of Convex Functions

Here we will talk about properties of convex (or concave upward) function.

We already noted that if function $$${f{{\left({x}\right)}}}$$$ is concave upward then $$$-{f{{\left({x}\right)}}}$$$ 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 $$$y={f{{\left({x}\right)}}}$$$ is defined and continuous on interval $$${X}$$$, and has finite derivative $$${f{'}}{\left({x}\right)}$$$ inside it. Function $$$y={f{{\left({x}\right)}}}$$$ is concave upward (downward) on $$${X}$$$ if and only if derivative $$${f{'}}{\left({x}\right)}$$$ is non-decreasing (non-increasing). Function is strictly concave upward (downward) on $$${X}$$$ if and only if derivative $$${f{'}}{\left({x}\right)}$$$ is increasing (decreasing).

Inflection Points

Definition. Point $$$c$$$ is an inflection point of function $$$y={f{{\left({x}\right)}}}$$$ if function at this point changes direction of concavity (i.e. from concave upward becomes concave downward or from concave downward becomes concave upward).