1
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.