What is the extreme f(x,y)=x^3+y^3-3x^2-6y^2-9x ?

The extreme f(x,y)=x^3+y^3-3x^2-6y^2-9x is Saddle(-1,4),Maximum(-1,0),Minimum(3,4),Saddle(3,0)

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extreme f(x,y)=x^3+y^3-3x^2-6y^2-9x

extreme points

f (x, y) = x^{3} + y^{3} - 3 x^{2} - 6 y^{2} - 9 x

Saddle (- 1, 4), Maximum (- 1, 0), Minimum (3, 4), Saddle (3, 0)

Solution steps

Find the critical points:

(- 1, 4), (- 1, 0), (3, 4), (3, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 6 y - 12

D (x, y) = (6 x - 6) (6 y - 12)

Check the sign of at each critical point

Check critical point

(- 1, 4) :

Saddle

Check critical point

(- 1, 0) :

Maximum

Check critical point

(3, 4) :

Minimum

Check critical point

(3, 0) :

Saddle

Saddle (- 1, 4), Maximum (- 1, 0), Minimum (3, 4), Saddle (3, 0)

h (x) = x^{3} - 12 x

f (x) = x^{2} - 11

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

\frac{2 ( x + 2 ) ^{2}}{x ^{2}}

2 x^{3} - 2 x + 6

What is the extreme f(x,y)=x^3+y^3-3x^2-6y^2-9x ?
The extreme f(x,y)=x^3+y^3-3x^2-6y^2-9x is Saddle(-1,4),Maximum(-1,0),Minimum(3,4),Saddle(3,0)