PM Mon Jun $30$

SCAICET$9$

$4\mathrm{.}2\mathrm{.}030.$MI$.$SA$.$

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Tutorial Exercise

Suppose that $3f^{\text{'}}(x)4$ for all values of $x.$ What are the minimum and maximum possible values of $f(8)-f(4)?$

Step $1$

Recall the mean value theorem, which states that if we let $f$ be a function that satisfies the following hypotheses, then there is a number $c$ in $(a,b)$ such that $f(b)-f(a)=f^{\text{'}}(c)(b-a).$

$f$ is continuous on the closed interval $a,b.$

$f$ is differentiable on the open interval $(a,b).$

We are interested in the minimum and maximum possible values of $f(8)-f(4).$ Therefore, in order for the mean value theorem to hold, we have the following for some $c$ in the interval $4,8.$

$f(b)-f(a)=f^{\text{'}}(c)(b-a)$

$f(8)-f(4)=f^{\text{'}}(c)(8-())$

$f(8)-f(4)=()f^{\text{'}}(c)$

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