Maximum value of $2$ variable algebric expression

Question

Finding maximum value of $f(a,b)=8a^3+27b^3-a^3b^3, a,b>0$

Trial: differentiate with respect to $a$ treating $b$ as a constant

So $f'(a,b)=24a^2-3a^2b^3$

differentiate with respect to $b$ treating $a$ as a constant

So $f'(a,b)=81b^2-3b^2a^3$

For finding maximum and minimum put $24a^2-3a^2b^3=0$ so either $a=0$ abd $b=2$ and also put $81b^2-3a^3b^2=0$

So we have $b=0$ and $a=8$

Could some help me how to find maximum of $f(a,b)$, thanks

Answer 1

In the situation that you gave, there is no maximum when $a,b>0$.

The function grows infinitely large for both large $a$ or $b$.