To check if $g(x,y)=x^4+y^2-10x^2y$ has a local minimum at $(0,0)$ or not

Question

I need to check if $g(x,y)=x^4+y^2-10x^2y$ has a local minimum at $(0,0)$ or not. The determinant of hessian matrix is zero, so that test is inconclusive. I observed the graph around origin but it's bit difficult to claim anything. What I tried is using two paths $y=x^2$ and $y=-x^2$. Using these two, $g$ simplifies to $-8x^4$ and $12x^4$, from which we can conclude in any neighbourhood of $0$, we can find points giving $g$ both positive and negative. So it is saddle point. Am I correct?

Answer 1

Yes, your reasoning is correct. Since the determinant of the Hessian matrix at $(0,0)$ is zero, the second derivative test is inconclusive. However, by considering the function along the paths $y=x^2$ and $y=-x^2$, you have shown that there are points in any neighborhood of $(0,0)$ that give both positive and negative values of $g(x,y)$. This implies that $(0,0)$ is a saddle point of $g(x,y)$, where the function neither has a local minimum nor a local maximum. Therefore, your conclusion that $(0,0)$ is a saddle point is correct.