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\noindent {\bf Question}

\noindent Show that the function $h(x) = \sqrt{x-1}$ satisfies the
hypotheses of the Mean Value Theorem on the interval $[2,5]$. Find
all the numbers $c$ in $(2,5)$ that satisfy the conclusion of the
Mean Value Theorem.

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\noindent {\bf Answer}

\noindent (Be sure to state the mean value theorem first, so that
it is clear that you know what the hypotheses and the conclusions
are.)  Note that $h(x)$ is continuous and differentiable on all of
$(1,\infty)$, since $x -1 >0$ on $x >1$, and so in particular $h$
is continuous on $[2,5]$ and differentiable on $(2,5)$ (i.e.,
satisfies the hypotheses).

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\noindent So, there exists some $c$ in $(2,5)$ at which
\[ h'(c) =\frac{h(5) -h(2)}{5-2} =\frac{1}{3}. \]
In fact, since $h'(c) = \frac{1}{2\sqrt{c -1}}$, the only solution
to $h'(c) =\frac{1}{3}$ occurs at $c = \frac{13}{4}$ (which does
lie in $(2,5)$, as expected).


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