Find the supremum, infimum, maximum and minimum of A

Question

A = {a ln a: a ∈ Q, a ≥ 0}.

I am stuck on this question. I started by letting f(a) = a ln a

Then solving for f'(a) = ln(a) + 1

Sup(A) = + infinity

I am not sure how to find Inf(A), Max(A), Min(A)

Answer 1

ln(a) + 1 = 0

ln(a) = -1

$a = e^{-1}$ would be an extrema but f is only defined on rationals. And the extrema would be $e^{-1}ln(e^{-1}) = -e^{-1}$

f''(a) = 1/a. So $f''(e^{-1}) = e > 0$ so there would be a local minimum at a = $e^{-1}$.

The is no point of minima but at all rational less than $e^{1} f are decreasing and at points higher it is increasing.

Although A is bounded below by $ -e^{-1}$, A has no minimum element.