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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.

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  • $\begingroup$ This appears to be known variously as Darboux's theorem. $\endgroup$ – Dan Rust Mar 9 '13 at 21:40
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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?

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Hint:
Define $h(x)= f(x)- \gamma \cdot x$. Then search for a maximum (which must exist on a compact interval).

Wlog assume $h'(a)>0$ and $h'(b)<0$ (if that is not the case, observe $-h(x)$ which fulfills the properties.

We see that although the derivative is not continuous in general, it still haves the intermediate value property (also known as darbaux property).

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  • $\begingroup$ A maximum is not sufficient, we need to look for an extreme point: for instance, $f(x) = x^2$ with $a = 0, b=1$ and $\gamma = 1$. $\endgroup$ – Pedro M. Mar 9 '13 at 21:53
  • $\begingroup$ @pedromilet yeah i assumed wlog $h'(a)>0$ and $h'(b)<0$ should have said that $\endgroup$ – Dominic Michaelis Mar 9 '13 at 21:56

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