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I am trying to find the maximum of $f(x,y)=(x+y)^4+y^4$ constrained to $x^4+y^4=1$.

Using Lagrange Multiplier I get $$ (x+y)^3=\lambda x^3 $$ $$ (x+y)^3+y^3=\lambda y^3 $$

But I don't see how to proceed after this.

Do you have some idea on this problem ?

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    $\begingroup$ Is $f(x,y)=x+y^4+y^4=x+2y^4$ right or do you have a typo ? $\endgroup$ – callculus Oct 21 '16 at 13:37
  • $\begingroup$ Thank you. I have corrected the formula. $\endgroup$ – Spout Oct 21 '16 at 13:45
  • $\begingroup$ Call $\mu$ the cube root of lambda to get that $x+y=\mu x$ and this should do. $\endgroup$ – Will M. Jan 19 '17 at 17:38
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I don't see a simple way to proceed from here, but the basic idea is that you now have three equations -- the two equations you derived and the constraint equation $x^4 + y^4 =1$ -- and three unknowns $x$, $y$, and $\lambda$. You then combine the three equations to solve for $x$ and $y$. (And you can also solve for $\lambda$ but that is just gravy.)

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