So I know the IVT says that given a continuous function on a closed interval [a,b] then if $f(a) < \gamma <f(b)$ then there is $c\in(a,b)$ such that $f(c) = \gamma$.
Is there any reason for not replacing the image of the end-points with the image of the maximum and minimum of $f$ on $[a,b]$. In other words, why isn't the mean value theorem stated as above but with $f(a)$ replaced by the minimum value of $f$ taken on $[a,b]$ and similarly $f(b)$ replaced with the maximum value of $f$ taken on $[a,b]$?