Find the maximum and minimum of $f(x,y)=xy-y+x-1$ on the set $x^2+y^2=2$

Question

So I started the problem by first finding the critical points using the partial derivatives, which turns out that there is only 1 critical point at $(1,-1)$ where $f(1,-1)=0$

Then I know I must look at the outside, or when $g(x,y)=x^2+y^2=2$, and I believe I could use $\nabla f=\lambda*\nabla g$, and this leaves me with the following equations

$$y+1=2x\lambda$$

$$x-1=2y\lambda$$

$$x^2+y^2=1$$

And I am unable to solve this set of equations. Could someone help me out on this step or see a flaw in an earlier step?

Answer 1

Divide the first two equations to eliminate $\lambda$. After clearing denominators, you have a quadratic equation in $x$ and $y$. In combination with the constraint, eliminate $x$ (or $y$). And then you are almost done.