For the two variable case above, we are checking $\det(\bm{H})$
The second derivative test based on the eigenvalues of the function’s Hessian matrix.
In particular, assuming that all second order partial derivatives of $f$ are continuous on a neighbourhood of a stationary point $\bm{x^\star}$ ,then if the eigenvalues of the Hessian at $\bm{x^\star}$ are all positive, i.e., if the Hessian is positive definite at $\bm{x^\star}$ ,then $\bm{x^\star}$ is a local minimum. If the eigenvalues are all negative, i.e., if the Hessian is negative definite at $\bm{x^\star}$ ,then $\bm{x^\star}$ is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second derivative test in inconclusive.
The constrains in the above question are