Prove that a Function is Concave

Suppose that f(x) has a continuous first derivative for all x is an element of R.

a) Prove that f(x) is concave if and only if f(x*) + (x-x*)f'(x*) >= f(x) for all x and x* is an element of R.

b) Given that f(x) is concave, prove that x* is a global maximum of f(x) if and only if f'(x*) 0.

c) Given that f(x) is strictly concave, prove that it cannot possess more than one global maximum.

Solution Summary

The solution lists the steps that are needed to get to the answer. The student can then use these steps and concepts and apply them to similar problems in the future. This solution is provided in an attached Word document.