If f$(2)=2$ and f $($x$)3$ for $2$ x $7,$ 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 $($c$)($b $$ a$).$ f is continuous on the closed interval $[$a$,$ b$].$ f is differentiable on the open interval $($a$,$ b$).$ Therefore, if the mean value theorem holds, we have the following for some c in $(2,7).$ f$($b$)$ f$($a$)=$f $($c$)($b $$ a$)$f$(7)$ f $=$f $($c$)(72)$