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))`:
|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
|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.