What is the extreme f(x,y)=x^2y+y^3-27y ?

The extreme f(x,y)=x^2y+y^3-27y is Minimum(0,3),Maximum(0,-3),Saddle(3sqrt(3),0),Saddle(-3sqrt(3),0)

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extreme f(x,y)=x^2y+y^3-27y

extreme points

f (x, y) = x^{2} y + y^{3} - 27 y

Minimum (0, 3), Maximum (0, - 3), Saddle (33, 0), Saddle (- 33, 0)

Solution steps

Find the critical points:

(0, 3), (0, - 3), (33, 0), (- 33, 0)

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

D (x, y) = 12 y^{2} - 4 x^{2}

Check the sign of at each critical point

Check critical point

(0, 3) :

Minimum

Check critical point

(0, - 3) :

Maximum

Check critical point

(33, 0) :

Saddle

Check critical point

(- 33, 0) :

Saddle

Minimum (0, 3), Maximum (0, - 3), Saddle (33, 0), Saddle (- 33, 0)

y = (x - 4)^{2} (x + 2)^{3}

f (x, y) = 2 x + y

f (x) = \frac{1}{3} x^{3} - x^{2} - 3 x + 3

f (x) = 5 x^{3} - 30 x^{2} + 45 x

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

What is the extreme f(x,y)=x^2y+y^3-27y ?
The extreme f(x,y)=x^2y+y^3-27y is Minimum(0,3),Maximum(0,-3),Saddle(3sqrt(3),0),Saddle(-3sqrt(3),0)