Question

find the critical values for f(x,y)=6xy^2 -2x^3 - 3y^4 test and classify the nature of the extrema

Answer 1

I recall the first several steps of this process, but will have to refer to a textbook (or to ask you to refer to your own) regarding the last part.

Find the 1st partial derivatives with respect to x and with respect to y, separately. Set these = to zero, and then solve the 2 resulting equations simultaeously for x and y.

Answer 2

Once you have your critical points, the next step is to apply the second partial derivative test. Compute the determinant of the Hessian matrix,
\[\det\begin{pmatrix}f_{xx}&f_{xy}\\[1ex]f_{yx}&f_{yy}\end{pmatrix}=f_{xx}f_{yy}-f_{xy}f_{yx}=f_{xx}f_{yy}-(f_{xy})^2\]where each partial derivative is evaluated at the critical point.

If the determinant and \(f_{xx}\) are both positive, then the critical point is a local minimum. If the determinant is positive and \(f_{xx}\) is negative, then the critical point is a maximum. If the determinant is negative, you have a saddle point. If it's zero, the test fails.