# Second derivative test for discriminant

Question: Find all critical points of $$f(x,y)=x^3+y^4-6x-2y^2$$

Apply 2nd Derivative test to each point and determine whether it is local maximum, local minimum or saddle point or that the test fails.

I have found six critical points in total and applying the discriminant equation for 2 variables [ discriminant= (fxx)(fyy) - (fxy)^2 , which is 24x(3y^2 -1) in this question ] , what should be my next step?

HINT

We have that for $$f(x,y)=x^3+y^4-6x-2y^2$$

• $$f_x=3x^2-6=0 \implies x=\pm \sqrt 2$$
• $$f_y=4y^3-4y=0 \implies y=0 \,\lor\,y=\pm 1$$

then evaluate

• $$f_{xx}=6x$$
• $$f_{xy}=f_{yx}=0$$
• $$f_{yy}=12y^2-4$$

at each critical point and consider the Hessian determinant at each point.

Refer to

• x should be equal to +/- root2? – CCola Dec 2 '18 at 14:38
• @CCola Opsss yes of course...I fix the typo! – user Dec 2 '18 at 14:41