Tutorial Exercise

If $f(4)=2$ and $f^{\text{'}}(x)2$ for $4x7,$ how small can $f(7)$ 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 $(4,7).$

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

$f(7)-f()=f^{\text{'}}(c)(7-4)$