Tutorial Exercise

If $f(3)=4$ and $f^{\text{'}}(x)2$ for $3x5,$ how small can $f(5)$ possibly be$?$

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 $(a,b).$

Therefore, if the mean value theorem holds, we have the following for some $c$ in $(3,5).$

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

$f(5)-f(,)=f^{\text{'}}(c)(5-3)$