Suppose $f : (0, 1) → \mathbb R$ is differentiable on $(0, 1)$ and that for $x,y ∈ (0,1)$ with $x < y$ we have that $f′(x) < 0$ and $f′(y) > 0$. Show that there exists $z ∈ (x, y)$ such that
$$f(z) = \inf\{f(a) : a \in [x,y]\}$$
I'm not really sure how to start this, I've looked at a few theorems like Rolle's and the mean value theorem but don't know which one to use?
Since $f$ is differentiable on $(0,1)$ it is continuous on $[x,y]$. Thus $f$ attains its infimum on $[x,y]$.
The conditions on the derivatives imply that the infimum can't occur at either endpoint.