It is slightly tricky to spot, but the core idea is the same as in the proof of the mean value theorem. So perhaps I should make this a leadup exercise.

The mean value theorem asserts we can find a c in (a,b) with f'(c)=(f(b)-f(a))/(b-a) if f is a differentiable function on [a,b].

B1. (Rolle's Theorem) Prove that if g is a differentiable function on [a,b] with g(a)=g(b), then g'(c)=0 for some c in (a,b). (Use the extreme value theorem to do this)

B2. (MVT) By choosing g appropriately show that f'(c)=(f(b)-f(a))/(b-a) for some c in (a,b). (The appropriate choice is easier to see here than in my original question, I think you are capable of finding it. We can also interpret this statement geometrically, which might help).

Then attempt the questions in my previous post. Aim to use Rolle's in a similar way to how it is used in Q2 of this post. With this target in mind, the choice of function should be easier for you to find.

## Bookmarks