Maximum value of $abc$ for $a, b, c > 0$ and $ab + bc + ca = 12$ $a,b,c$ are three positive real numbers such that $ab+bc+ca=12$.
Then find the maximum value of $abc$

Or, the same answer also comes by intuition. since a,b,c are positive numbers, $(ab).(bc).(ca)$ is maximum only when $ab=bc=ca=\frac{12}{3}=4$. therefore, $ abc={((ab)(bc)(ca))}^\frac{1}{2}=4^\frac{3}{2}=8$.
