# Contest Math: Finding maximum value under restrictions

$p,q,r,s$ 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.

• I don't know how to use LaTeX, I'm a beginner. If you can invest your precious time in editing the question, it'd be great. Thanks. Could you please help in solving the problem instead, though? – arya_stark Dec 28 '17 at 5:56

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.