Answer
See explanations.
Work Step by Step
Step 1. The middle term of the inequalities is half of the statement in the Mean Value Theorem where we have $f'(c)=\frac{f(b)-f(a)}{b-a}$
Step 2. For the value of $f'(c)$ to lie in between the minimum and maximum of $f'(x)$, we need to avoid the situation that $f'(c)$ is a point of discontinuity (which could be outside the range).
Step 3. We conclude that the requirement for the inequalities to be true is that $f'(x)$ should be a continuous function.
