Question

Find the maximum value of $x^m{y}^{m}wherem,n>0$ such that x+y =a

Answer 1

$x+y=a⟹m(\frac{x}{m})+n(\frac{y}{n})=a$

$AM\ge GM⟹\frac{m(\frac{x}{m})+n(\frac{y}{n})}{m+n}\ge \sqrt[m+n]{m+n(\frac{x}{m}{)}^m(\frac{y}{n}{)}^n}$

$x^m{y}^n\le \frac{a^{m+n}.m^m.n^n}{(m+n{)}^{m+n}}$

The maximum value is $\frac{a^{m+n}.m^m.n^n}{(m+n{)}^{m+n}}$