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.

Property 1. Suppose $$${f{{\left({x}\right)}}}$$$ is concave upward and $$${c}>{0}$$$ is arbitrary constant then $$${c}{f{{\left({x}\right)}}}$$$ is concave upward.

Property 2. Sum of any number of concave upward functions is concave upward.

Note, that this doesn't hold for product, i.e. product of two concave upward functions can be not concave upward.

Property 3. Suppose that we are given two functions $$${y}={f{{\left({u}\right)}}}$$$ and $$${u}={g{{\left({x}\right)}}}$$$. Following is true for composite function $$${f{{\left({g{{\left({x}\right)}}}\right)}}}$$$:

$$${f{{\left({u}\right)}}}$$$	$$${g{{\left({x}\right)}}}$$$	$$${f{{\left({g{{\left({x}\right)}}}\right)}}}$$$
concave upward, increasing	concave upward	concave upward
concave upward, decreasing	concave downward	concave upward
concave downward, increasing	concave downward	concave downward
concave downward, decreasing	concave upward	concave downward

Property 4. If for the function $$${y}={f{{\left({x}\right)}}}$$$ there exists unique inverse $$${{f}}^{{-{1}}}{\left({x}\right)}$$$ then the following is true

$$${f{{\left({x}\right)}}}$$$	$$${{f}}^{{-{1}}}{\left({x}\right)}$$$
concave upward, increasing	concave downward, increasing
concave upward, decreasing	concave upward, decreasing
concave downward, increasing	concave upward, increasing
concave downward, decreasing	concave downward, decreasing

Property 5. Non-constant concave upward on interval $$${X}$$$ function $$${f{{\left({x}\right)}}}$$$ can't attain global maximum inside this interval. In other words global maximum can be only at one of the endpoints.