Let $f(x,y)=\frac {x^2}{2}+\frac {y^2}{4}$ on $\{(x,y)|x^2-y^2=2\}$. Find the absolute maximum and minimum if they exist.

Question

Let $f(x,y)=\frac {x^2}{2}+\frac {y^2}{4}$ on $\{(x,y)|x^2-y^2=2\}$. Find the absolute maximum and minimum if they exist.

I approached this problem using lagrange multiplier, with $g(x,y)=x^2-y^2-2$.

$f_x=x, f_y=y$ and $g_x=2x, g_y=-2y$. Setting these equal, I got $1=2\lambda$ and $-1=2\lambda$, which has no solution.

Can someone please tell me where I did wrong?

Thanks!

Answer 1

To answer the actual question, you should be getting three equalities:

$$ \left\{ \begin{array}{rl} x-2\lambda x & = 0, \\ y/2+2\lambda y & =0, \\ x^2-y^2 & =2 \end{array} \right.$$

So you know that $y=0$ and hence that $x=\sqrt{2}$.