Let $f$ be continuous and differentiable on the interval $[a, b]$. Assuming $f$ is bounded on the interval $[a, b]$ and $m = \inf\limits_{[a,b]} f(x)$, prove that there exists $d \in [a, b]$ such that $f(d) = m$. You can use the fact that a function which is continuous on a closed interval is bounded.
I reckon this has something to do with that mean value theorem but im not entirey sure how to do this. Any help would be appreciated.
Many thanks.
Since $f$ is continuous at $[a,b]$ which is compact then $f$ attains his minimum at some point $\xi$.