First find all critical points which are the points \( (x,y) \) which solve, $$ \begin{cases} f_x(x,y) = 0 \\ f_y(x,y) = 0 \end{cases} $$ simultaneously.

Next compute all second order partial derivatives, \(f_{xx},f_{yy}\), and \(f_{xy} \).

For each critical point, \( (a,b) \), evaluate the Hessian determinant applied to each critical point, $$ D = f_{xx}(a,b) f_{yy}(a,b) - f_{xy}(a,b)^2 $$

Then you can classify the critical point according to the following rules,

1. If \(D> 0\) and \(f_{xx}(a,b) >0 \) then \((a,b)\) is a local minimum,
2. If \(D > 0\) and \(f_{xx}(a,b) < 0 \) then \((a,b)\) is a local maximum,
3. If \(D < 0\) then \((a,b)\) is a saddle-point,
4. If \(D = 0\) then the second-derivative test failed.

First we can compute the partial derivatives of \(f\) and set them equal to zero,

We can simplify these equations to become

We can solve these equations to find the only critical point,

Next, we have to compute all second order partial derivatives of \(f\),

Of course this then gives,