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