What is the extreme f(x,y)=x^3+y^3+3x^2-18y^2+81y+5 ?

The extreme f(x,y)=x^3+y^3+3x^2-18y^2+81y+5 is Minimum(0,9),Saddle(0,3),Saddle(-2,9),Maximum(-2,3)

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extreme f(x,y)=x^3+y^3+3x^2-18y^2+81y+5

extreme points

f (x, y) = x^{3} + y^{3} + 3 x^{2} - 18 y^{2} + 81 y + 5

Minimum (0, 9), Saddle (0, 3), Saddle (- 2, 9), Maximum (- 2, 3)

Solution steps

Find the critical points:

(0, 9), (0, 3), (- 2, 9), (- 2, 3)

\frac{\partial ^{2} f}{\partial x ^{2}} = 6 y - 36

D (x, y) = (6 x + 6) (6 y - 36)

Check the sign of at each critical point

Check critical point

(0, 9) :

Minimum

Check critical point

(0, 3) :

Saddle

Check critical point

(- 2, 9) :

Saddle

Check critical point

(- 2, 3) :

Maximum

Minimum (0, 9), Saddle (0, 3), Saddle (- 2, 9), Maximum (- 2, 3)

f (x, y) = x^{3} + y^{3} + 3 y^{2} - 3 x - 9 y + 2

f (x, y) = x^{4} + 3 x y^{3} - x y

f (x) = 3 x^{2} - 3

f (x, y) = 1 - x - y

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

What is the extreme f(x,y)=x^3+y^3+3x^2-18y^2+81y+5 ?
The extreme f(x,y)=x^3+y^3+3x^2-18y^2+81y+5 is Minimum(0,9),Saddle(0,3),Saddle(-2,9),Maximum(-2,3)