# Properties of Convex Functions

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

We already noted that if function f(x) is concave upward then -f(x) is concave downward. So, these properties also hold for concave downward functions.

Property 1. Suppose f(x) is concave upward and c>0 is arbitrary constant then cf(x) 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(u) and u=g(x). Following is true for composite function f(g(x)):

 f(u) g(x) f(g(x)) 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(x) there exists unique inverse f^(-1)(x) then the following is true

 f(x) f^(-1)(x) 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(x) can't attain global maximum inside this interval. In other words global maximum can be only at one of the endpoints.