Find the critical points. (Multivariable Calculus) It seems like an easy calcula

Question

Find the critical points. (Multivariable Calculus)

It seems like an easy calculation. Why is the top question only 50% correct?

Entered Answer Preview Result 50% correct minimum, none incorrect minimum, none, maximum, none minimum, none, maximum, none t least one of the answers above is NOT correct 1 pt) Find the critical points for the function and classify each as a local maximum, local minimum, saddle point, or none of these critical points: (-2,-1), (-2,1), (0,1), (0,-1) (give your points as a comma separated list of (x,y) coordinates.) classifications: minimum, none, maximum, none (give your answers in a comma separated list, specifying maximum, minimum, saddle point, or none for each, in the same order as you entered your critical points)

Explanation / Answer

f(x,y)=x3+y3-3x2-3y -2

for critical points fx=0,fy=0

fx=3x2-6x=0,fy=3y2-3 =0

3x(x-2)=0 ,3(y2-1)=0

x=0,x=2 ,y =-1,y=1

critical points are (x,y)=(0,-1),(0,1),(2,-1),(2,1)

fxx=6x-6,fyy=6y,fxy=0, D=fxxfyy-fxy2

(x,y)=(0,-1) => fxx=-6,fyy=-6,fxy=0, D=36>0, fxx<0 =>local maximum

(x,y)=(0,1) => fxx=-6,fyy=6,fxy=0, D=-36<0 =>saddle points

(x,y)=(2,-1) => fxx=6,fyy=-6,fxy=0,D=-36<0 =>saddle points

(x,y)=(2,1) => fxx=6,fyy=6,fxy=0,D=36>0, fxx>0 =>local minimum

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