# Convex functions and increasing derivatives

• Nov 10th 2010, 08:25 PM
Zennie
Convex functions and increasing derivatives
Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing.

Not really sure how to relate the two. The second part of the question asks that it then be assumed that f" exists and to show that f is convex if and only if f" exists and is greater than or equal to 0.

How do I show that f is convex iff f' is increasing before using f"? Explanation would be very helpful.
• Nov 10th 2010, 08:54 PM
Drexel28
Quote:

Originally Posted by Zennie
Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f' is monotonically increasing.

Not really sure how to relate the two. The second part of the question asks that it then be assumed that f" exists and to show that f is convex if and only if f" exists and is greater than or equal to 0.

How do I show that f is convex iff f' is increasing before using f"? Explanation would be very helpful.

This sounds like a Rudin problem.

Are you aware of the fact that $f$ is convex if and only if the function $\phi(x,y)=\frac{f(x)-f(y)}{x-y}$ is monontonically nondecreasing for fixed $y$?