What is the extreme f(x,y)=(x^2-9)^2+(y^2-36)^2 ?

The extreme f(x,y)=(x^2-9)^2+(y^2-36)^2 is

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extreme f(x,y)=(x^2-9)^2+(y^2-36)^2

extreme points

f (x, y) = (x^{2} - 9)^{2} + (y^{2} - 36)^{2}

Maximum (0, 0), Saddle (0, - 6), Saddle (0, 6), Saddle (- 3, 0), Minimum (- 3, - 6), Minimum (- 3, 6), Saddle (3, 0), Minimum (3, - 6), Minimum (3, 6)

Solution steps

Find the critical points:

(0, 0), (0, - 6), (0, 6), (- 3, 0), (- 3, - 6), (- 3, 6), (3, 0), (3, - 6), (3, 6)

\frac{\partial ^{2} f}{\partial x ^{2}} = 12 y^{2} - 144

D (x, y) = (12 x^{2} - 36) (12 y^{2} - 144)

Check the sign of at each critical point

Check critical point

(0, 0) :

Maximum

Check critical point

(0, - 6) :

Saddle

Check critical point

(0, 6) :

Saddle

Check critical point

(- 3, 0) :

Saddle

Check critical point

(- 3, - 6) :

Minimum

Check critical point

(- 3, 6) :

Minimum

Check critical point

(3, 0) :

Saddle

Check critical point

(3, - 6) :

Minimum

Check critical point

(3, 6) :

Minimum

Maximum (0, 0), Saddle (0, - 6), Saddle (0, 6), Saddle (- 3, 0), Minimum (- 3, - 6), Minimum (- 3, 6), Saddle (3, 0), Minimum (3, - 6), Minimum (3, 6)

f (x) = - 3 x^{2} - 12 x + 24 y^{2} + 12 y = 0

f (x) = 7 x - 5

ln (x) - 7 x + 11

f (x, y) = 3 - x^{2} - y^{2} + 6 y

f (x) = 1 - 1

What is the extreme f(x,y)=(x^2-9)^2+(y^2-36)^2 ?
The extreme f(x,y)=(x^2-9)^2+(y^2-36)^2 is