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?



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

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

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