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)

f (x) = x^{4} - 2 x^{2} + 5, - 2 \leq x \leq 2

\frac{x ^{\frac{1}{3}}}{x ^{2} + 8} - 10

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

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

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

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)