I need help to find critical points of the function:
$$f(x,y)=\frac{-x^3}{3}+x-y^2$$
Then I have to classify these critical points as local maxima/minima or saddle points.
I thought that to find the critical points, I have to find the 1st derivative and to find local max/min or saddle, I have to use the second derivative test. I am having a little trouble both in finding first and second derivatives and how to use it to find the given above. Can someone help me?
Edit: I found the critical points to be $(1,0)$ and $(-1,0)$. Can someone verify this as well?
Best Answer
You have $$\nabla f(x,y) = \begin{bmatrix} -x^2 + 1 \\ - 2y \end{bmatrix}.$$ So the critical points are $(-1,0)$ and $(1,0)$.
Now, the Hessian is $$\nabla^2 f(x,y) = \begin{bmatrix} -2x & 0 \\ 0 & -2 \end{bmatrix}.$$ The eigenvalues of $\nabla^2 f(-1,0)$ are $2$ and $-2$. Thus, $\nabla^2 f(-1,0)$ is indefinite and $(-1,0)$ is a saddle point. The eigenvalues of $\nabla^2 f(1,0)$ are $-2$ and $-2$. Thus, $\nabla^2 f(1,0)$ is negative definite and $(1,0)$ is a maximum.