Intermediate Value Theorem and Continuity of derivative. Suppose that a function $f(x)$ is differentiable $\forall x \in [a,b]$. Prove that $f'(x)$ takes on every value between $f'(a)$ and $f'(b)$.
If the above question is a misprint and wants to say "prove that $f(x)$ takes on every value between $f(a)$ and $f(b)$", then I have no problem using the intermediate value theorem here.
If, on the other hand, it is not a misprint, then it seems to me that I can't use the Intermediate value theorem, as I can't see how I am authorised to assume that $f'(x)$ is continuous on $[a,b]$.
Or perhaps there is another way to look at the problem?

This is not a misprint. You can indeed prove that $f'$ takes every value between $f'(a)$ and $f'(b)$. You cannot, however, assume that $f'$ is continuous. A standard example is $f(x) = x^2 \sin(1/x)$ when $x \ne 0$, and $0$ otherwise. This function is differentiable at $0$ but the derivative isn't continuous at it.
To prove the problem you have, consider the function $g(x) = f(x) - \lambda x$ for any $\lambda \in (f'(a), f'(b))$. What do you know about $g'(a)$, $g'(b)$? What do you conclude about $g$ in the interval $[a, b]$?
