I have an example of a set
I decided to use $f(a) = a + (2a)^{-1} \implies f'=1 -2(2a)^{-2} = 1-1/a^2=0 \iff a=\frac{\pm1}{\sqrt{2}}$
Now I realize the critical values aren't even in the set, but taking $f(a)$ at those values anyways I get
$f(1/\sqrt{2}) =\sqrt{2}$
$f(-1/\sqrt{2}) =0$
Now evaluating through the end points, we get
$f(0.1) = 51/10$
$f(5) = 51/10$
So although the critical values don't belong to $\mathbb{Q}$, we do get that
sup(B) = 51/10
inf(B) = 0
max(B)=min(B) = empty