extreme values $xy^2$ on unit disc/unit circle

Question

I am trying to find the local and global extreme values for $f(x,y)=xy^2$.

I tried the Lagrange Multiplicator method. From my understanding $$ \nabla_f=\nabla_g \lambda, $$ with $\nabla_g$ beeing my secondary condition, which is in my case either (x,y) beeing on the unit circle, or (x,y) beeing on the open unit disc (I want to do both).

I tried it with x,y beeing on the unit circle as my secondary condition ($x^2 + y^2 = 1$) an got $$\left\{\begin{array}{l}y^2=2\lambda x\\2xy=2\lambda y\\x^2+y^2=1\end{array}\right.$$

But I feel kind of stuck now. Obviously $x= \pm 1, y=0, \lambda = 0$ is a solution. I fail to find another solution.

I am also not sure how to implement the other secondary condition, (x,y) being on the open unit disc.

Answer 1

Guide:

• If $y=0$, then $x^2=1$ and hence $\lambda = 0$.

• If $y \ne 0$, then $\lambda = x$, hence hence $y^2=2x^2$. We have $3x^2=1$.

Can you complete the task now?