Application of mean value theorem

Question

$f$ is differentiable in $[a,b]$, and satisfies $f(a)<f(b)$, $f'(a)=f'(b)=0$. Show that there exists $c$ in $[a,b]$

\begin{equation} \frac{f(c)-f(a)}{c-a}=f'(c) \end{equation}

How is the mean value theorem \begin{equation} \frac{f(b)-f(a)}{b-a}=f'(c) \end{equation} applicable here?

This is Flett's Mean Value Theorem. Mean Value Theorem is applied in the proof in the form of Rolle's Theorem. This link may help: www.math.sc.edu/~girardi/m555/current/hw/MVT-Flett.pdf — user71352, Aug 01 '13 at 02:33
Your question appears to be simply a special case of the mean value theorem. Am I missing something? Where are you stuck? (Argh sorry I read the $c$'s as $b$'s in the first equation) — Hugh, Aug 01 '13 at 02:35
@use71352 Thank you very much for posting the link that is very helpful. — stph, Aug 02 '13 at 03:07

score 5 · Accepted Answer · edited Nov 27 '19 at 04:56

Consider the auxiliary function $$g(x):={f(x)-f(a)\over x-a} \quad(a<x\leq b), \quad g(a):=0\ .$$ Then $g$ is continuous on $[a,b]$ and differentiable in $(a,b)$. One computes $$g'(x)={(x-a) f'(x)-\bigl(f(x)-f(a)\bigr)\over (x-a)^2}\qquad(0<x\leq b)\ .$$ If $g(b)=0$, then this is a consequence of Rolle's Theorem.

If $g(b)>0$, then $g'(b)<0$; therefore $g(x)>g(b)>0$ immediately to the left of $b$. This allows us to conclude that $g$ assumes its maximum on $[0,b]$ in an interior point $c$. From $g'(c)=0$ it then follows that $${f(c)-f(a)\over c-a}=f'(c)\ .$$

The case for $g(b)<0$ is similar.

Thank you very much Mr. Blatter, for your answer. – stph Aug 02 '13 at 00:21 — stph, Aug 02 '13 at 00:21

score 4 · Answer 2 · answered Aug 01 '13 at 20:03

4

As mentioned in the comments, this is Flett's version of the MVT. A proof of this can be found at this link, also from the comments.

answered Aug 01 '13 at 20:03

apnorton

17,706

In regards to the link, what is Flett reverencing when he says "then there exists an $x_1$ in $a<x_1<b$ such that $\phi(x_1)>\phi(b)$"? – CarlSagan Nov 27 '19 at 04:34

Application of mean value theorem

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