Find the local maximum and minimum values and saddle point(s) of the function.

Critical points occur where the gradient is zero. This is guaranteed whenever and either or .

The Hessian matrix for this function looks like



and has determinant



Maxima occur whenever the determinant is positive and . Minima occur whenever both the determinant and are positive. Saddle points occur whenever the determinant is negative.

At , you have a saddle point since the determinant reduces to -324, so is the saddle point.

At , the determinant is and , so is a local maximum.

No other critical points remain, so you're done.