What is the extreme y^3+3x^2y-6x^2-6y^2+2 ?

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

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extreme y^3+3x^2y-6x^2-6y^2+2

extreme points

y^{3} + 3 x^{2} y - 6 x^{2} - 6 y^{2} + 2

Minimum (4, 0), Maximum (0, 0), Saddle (2, 2), Saddle (2, - 2)

Solution steps

Find the critical points:

(4, 0), (0, 0), (2, 2), (2, - 2)

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

D (y, x) = (6 y - 12)^{2} - 36 x^{2}

Check the sign of at each critical point

Check critical point

(4, 0) :

Minimum

Check critical point

(0, 0) :

Maximum

Check critical point

(2, 2) :

Saddle

Check critical point

(2, - 2) :

Saddle

Minimum (4, 0), Maximum (0, 0), Saddle (2, 2), Saddle (2, - 2)

f (x) = 3000 x^{2} - 2000 x + 8000

f (x) = x^{3} - 5 x^{2} + 6 x

\frac{x}{x ^{2} + 49}

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

f (x) = x e^{\frac{x}{3}}

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