Proving an inequality usuing Lagrange multipliers

Question

Cant seem to find the trick, how to proove this one, usuing Lagrange's amazing multipliers.

$$ \bigg(\frac{x+y}{2}\bigg)^n \le \frac{x^n + y^n}{2} $$

$ x,y > 0$, and $n\in \Bbb N$.

Any tips?

Thanks guys!

Answer 1

Try minimizing $f(x,y) = (\frac{x^n+y^n}{2})$ subject to the constraint $(\frac{x+y}{2})^n = C$, where $C>0$ is a constant. You should be able to show that when the minimum occurs, $x=y$, which will give you the result you want.