Problem. Is it true that $h(x) = f(g(x))$ has global maximum or global minimum if $g$ is continuous in $[a,b]$, $f$ is continuous in $[c,d]$ with the image of $f$ contained in $[a,b]$?
Attempt. I say no because it's not clear that $h(x)$ is well-defined. Nothing is said about the image of $g$. For $f$ to be well-defined, the domain of $f$ must be the same as the image of $g$. Without a guarantee that $f$ is well-defined, $h$ has no guarantee either, so we cannot use the extreme value theorem here.