What is the extreme f(x,y)=-(x^2-1)^2-(x^2y-x-1)^2 ?

The extreme f(x,y)=-(x^2-1)^2-(x^2y-x-1)^2 is Maximum(1,2),Maximum(-1,0)

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extreme f(x,y)=-(x^2-1)^2-(x^2y-x-1)^2

extreme points

f (x, y) = - (x^{2} - 1)^{2} - (x^{2} y - x - 1)^{2}

Maximum (1, 2), Maximum (- 1, 0)

Solution steps

Find the critical points:

(1, 2), (- 1, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 2 x^{4}

D (x, y) = - 40 x^{6} y^{2} + 24 x^{6} + 72 x^{5} y + 56 x^{4} y - 40 x^{4} - 48 x^{3} - 16 x^{2}

Check the sign of at each critical point

Check critical point

(1, 2) :

Maximum

Check critical point

(- 1, 0) :

Maximum

Maximum (1, 2), Maximum (- 1, 0)

0.5 x^{2} + 15 x + 5000

f (x) = x^{3} - x - 1

f (x) = \frac{2 x ^{2} + 6}{x - 1}

f (x) = x^{5} + 32

f (x) = (x + 1)^{2} (x - \frac{1}{2}) (x - 2)

What is the extreme f(x,y)=-(x^2-1)^2-(x^2y-x-1)^2 ?
The extreme f(x,y)=-(x^2-1)^2-(x^2y-x-1)^2 is Maximum(1,2),Maximum(-1,0)