$p,q,r,s$ 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.

  • $\begingroup$ 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? $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.