Let $f : (0,1) → R$ be continuous on $(0,1)$ with
$\lim_{x\to 0} f(x) = \lim_{x\to 1} f(x) = 0$
and where $f(x) > 0$ for all $x ∈ (0,1)$.
Show that:
(a) there exists $z ∈ (0,1)$ such that $f(z)$ = sup {$f(x) : x ∈ (0,1)$},
(b) there does not exist $z ∈ (0,1)$ such that $f(z)$ = inf {$f(x) : x ∈ (0, 1)$}.
I can visualise this in my mind but can't figure out where to start.