What is the extreme f(x,y)=2x^3+9xy^2+15x^2+27y^2 ?

The extreme f(x,y)=2x^3+9xy^2+15x^2+27y^2 is Minimum(0,0),Maximum(-5,0),Saddle(-3,2),Saddle(-3,-2)

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extreme f(x,y)=2x^3+9xy^2+15x^2+27y^2

extreme points

f (x, y) = 2 x^{3} + 9 x y^{2} + 15 x^{2} + 27 y^{2}

Minimum (0, 0), Maximum (- 5, 0), Saddle (- 3, 2), Saddle (- 3, - 2)

Solution steps

Find the critical points:

(0, 0), (- 5, 0), (- 3, 2), (- 3, - 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = 18 x + 54

D (x, y) = (12 x + 30) (18 x + 54) - 324 y^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- 5, 0) :

Maximum

Check critical point

(- 3, 2) :

Saddle

Check critical point

(- 3, - 2) :

Saddle

Minimum (0, 0), Maximum (- 5, 0), Saddle (- 3, 2), Saddle (- 3, - 2)

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

f (x) = - \frac{1}{3} x^{3} + 2 x^{2} - 3 x - 12

f (x) = \frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 6 x + 1

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

f (x, y) = 4 x^{3} + y^{3} - 12 x - 3 y

What is the extreme f(x,y)=2x^3+9xy^2+15x^2+27y^2 ?
The extreme f(x,y)=2x^3+9xy^2+15x^2+27y^2 is Minimum(0,0),Maximum(-5,0),Saddle(-3,2),Saddle(-3,-2)