What is the extreme 4x^2+4y^2+x^4+y^4-6x^2y^2 ?

The extreme 4x^2+4y^2+x^4+y^4-6x^2y^2 is

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extreme 4x^2+4y^2+x^4+y^4-6x^2y^2

extreme points

4 x^{2} + 4 y^{2} + x^{4} + y^{4} - 6 x^{2} y^{2}

Minimum (0, 0), Saddle (- 1, 1), Saddle (1, 1), Saddle (- 1, - 1), Saddle (1, - 1), Saddle (- 1, 1), Saddle (1, 1), Saddle (- 1, - 1), Saddle (1, - 1)

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 12 y^{2} + 8 - 12 x^{2}

D (x, y) = - 144 x^{4} - 288 x^{2} y^{2} - 144 y^{4} + 64

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- 1, 1) :

Saddle

Check critical point

(1, 1) :

Saddle

Check critical point

(- 1, - 1) :

Saddle

Check critical point

(1, - 1) :

Saddle

Check critical point

(- 1, 1) :

Saddle

Check critical point

(1, 1) :

Saddle

Check critical point

(- 1, - 1) :

Saddle

Check critical point

(1, - 1) :

Saddle

Minimum (0, 0), Saddle (- 1, 1), Saddle (1, 1), Saddle (- 1, - 1), Saddle (1, - 1), Saddle (- 1, 1), Saddle (1, 1), Saddle (- 1, - 1), Saddle (1, - 1)

y = \frac{x ^{2} - 6 x + 9}{( x - 1 ) ^{2}}

f (x, y) = x^{2} + 4 y^{2} + 1

f (x) = x^{2} + y^{2} - x y + x + y

f (x) = \frac{x}{1 - x}

f (x) = 2 x + 4 y

What is the extreme 4x^2+4y^2+x^4+y^4-6x^2y^2 ?
The extreme 4x^2+4y^2+x^4+y^4-6x^2y^2 is