(a bit difficult) finding maximum and minimum maybe using Lagrange's multiplier

Question

thanks for reading this first :D

I'm trying to solve the problem

"Finding the maximum and minimum of

$x^2+2y$

under the conditions $x^3+3xy+y^3=5, y\geq0$"

I would super appriciate your help. Cheers.

Answer 1

I'm giving you some hint to get proceed. Use Lagrange multiplier.

The two equations will be $ 2x=k(3x^2+3y) $ and $2= k(3x+3y^2)$. From 1st equation put $k=0$ , you can easily check this holds no good as it demands $2=0$

Next divide those equations. You will get conditions like $y=0$ and $xy=1$. From $y=0$, you will immediately get a value of $x$.

On the equation $x^3+3xy+y^3=5$, put $xy=1$. You get solutions like $x+y=2$. And I'm sure you can proceed thereafter.