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.