$p,q,r,s$ are non negative real numbers.
$p^5 + q^5\leq 1$ and $r^5+ s^5 \leq 1$
Find the maximum value of $p^2r^3 + q^2s^3$
I thought of using Holder's Inequality, but couldn't get to any specific maximum value of the expression.
Of course, using Lagrange Multipliers is a method but not a good one (it's cumbersome)
Could someone please give a detailed solution to the problem? Thanks a lot.
Using Holder is straightforward, $$1 \geqslant (p^5+q^5)^{2/5} \cdot (r^5+s^5)^{3/5} \geqslant (p^2r^3+q^2s^3)$$
Equality is possible when $p=q=r=s=\frac1{\sqrt[5]2}$, so that's the maximum. Any details you need, you should ask for.