Let $a,b \in \mathbb{R}$, $a<b$ and let $f$ be a differentiable real-valued function on an open subset of $\mathbb{R}$ that contains [a,b]. Show that if $\gamma$ is any real number between $f'(a)$ and $f'(b)$ then there exists a number $c\in(a,b)$ such that $\gamma=f'(c)$.
Hint: Combine mean value theorem with the intermediate value theorem for the function $\frac{(f(x_1)-f(x_2))}{x_1-x_2}$ on the set $\{(x_1,x_2)\in E^2: a\leq x_1 < x_2 \leq b\}$.
This is question number 7 on page 109 of Rosenlicht (introduction to Analysis).
I am having a lot of trouble trying to start on this problem.
Best Answer
Hint: Let $$g(x_1,x_2) = \frac{f(x_1) - f(x_2)}{x_1 - x_2}$$
Then $$\lim_{x_2 \searrow a} g(a,x_2) = f'(a) \text{ and } \lim_{x_1 \nearrow b} g(x_1,b) = f'(b)$$
So if $\varepsilon$ is sufficiently small, what can you say about $g(a,a+\varepsilon), \gamma$ and $g(b-\varepsilon,b)$?
Now if you define $h(t) = g(a + (b-\epsilon-a)t, a+\epsilon + (b-a-\epsilon)t)$, what does the Intermediate Value Theorem tell you?