Given that $a,b$ are both postive real numbers that sums to a constant $c$, find the maximum value of $a^{b}$.
Attempt:
$a+b=c$
$b=a-c$
$a^{b}=a^{c-a}$
$\frac{d}{da}=a^{c-a-1}(c-a)-a^{c-a}\ln{a}$
It seems to be impossible to find a neat solution for $a$ in this case, let alone $a^b$.
Is there another way that I could compute the maxima for $a^{b}$?