$f(x,y)=(4y^2 −x^2)e^{−x^2−y^2}$ on the domain $x^2 + y^2 \le 1$
Find extreme values of $f(x,y)$ given the constraint $g(x,y)=x^2 + y^2 = 1$
I have found $\nabla f$ and set it equal to $\lambda \nabla g$. What should I be solving for after this? Any help is appreciated.
$$\lambda = -e^{-x^2 - y^2} - e^{-x^2 - y^2}(4y^2 -x^2)$$ and $$\lambda = 4e^{-x^2 - y^2} - e^{-x^2 - y^2}(4y^2 -x^2).$$
Just substitute $x^2 = 1-y^2$ in the function to get that along the boundary of the region you have
$$g(y) = f(1-y^2,y) = (5y^2 - 1)e^{-1}$$
Now it's not hard to conclude that the minimum occurs when $y=0$ and the maximum when $y=\pm 1$ and they are $-e^{-1}$ and $4e^{-1}$, respectively.