Second derivative test for a function of two variables

Question

Suppose $f$ is a function of two variables $x,y$.

Suppose $(x_0,y_0)$ is a point in the domain of $f$ such that both the first-order partial derivatives at $(x_0,y_0)$ are zero, i.e., $f_x(x_0,y_0) = f_y(x_0,y_0) = 0$.

Now, I want to decide if this point $(x_0,y_0)$ is max.

Is it enough if I show that $f_{xx}(x_0,y_0) < 0$ and $f_{yy}(x_0,y_0) < 0$ without calculating the Hessian determinant and why?

Thanks

Answer 1

No, consider $f(x,y) = xy - (x-y)^2.$ We have $f(0,0)=0,$ and $f_{xx}(0,0) = -2 = f_{yy}(0,0).$ But $f(x,x) =x^2.$ So $(0,0)$ is not a maximum.