$p,q,r,s$ 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.