What is the extreme f(x,y)=2x^3-6x^2+y^3+21y^2 ?

The extreme f(x,y)=2x^3-6x^2+y^3+21y^2 is Saddle(0,0),Maximum(0,-14),Minimum(2,0),Saddle(2,-14)

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extreme f(x,y)=2x^3-6x^2+y^3+21y^2

extreme points

f (x, y) = 2 x^{3} - 6 x^{2} + y^{3} + 21 y^{2}

Saddle (0, 0), Maximum (0, - 14), Minimum (2, 0), Saddle (2, - 14)

Solution steps

Find the critical points:

(0, 0), (0, - 14), (2, 0), (2, - 14)

\frac{\partial ^{2} f}{\partial x ^{2}} = 6 y + 42

D (x, y) = (6 y + 42) (12 x - 12)

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(0, - 14) :

Maximum

Check critical point

(2, 0) :

Minimum

Check critical point

(2, - 14) :

Saddle

Saddle (0, 0), Maximum (0, - 14), Minimum (2, 0), Saddle (2, - 14)

x^{2} - 4 x + 3

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

f (x, y) = 25 - x^{2} - y^{2}

- 7 (x + 3)^{2} (x - 1) (x - 5)

f (x) = 2 x^{2} - 5 x y + 3 y^{4} + 5

What is the extreme f(x,y)=2x^3-6x^2+y^3+21y^2 ?
The extreme f(x,y)=2x^3-6x^2+y^3+21y^2 is Saddle(0,0),Maximum(0,-14),Minimum(2,0),Saddle(2,-14)