What is the extreme F(x,y)=x^3+3xy^2-3y^2-3x^2+11 ?

The extreme F(x,y)=x^3+3xy^2-3y^2-3x^2+11 is Minimum(2,0),Maximum(0,0),Saddle(1,1),Saddle(1,-1)

Popular >

extreme F(x,y)=x^3+3xy^2-3y^2-3x^2+11

extreme points

F (x, y) = x^{3} + 3 x y^{2} - 3 y^{2} - 3 x^{2} + 11

Minimum (2, 0), Maximum (0, 0), Saddle (1, 1), Saddle (1, - 1)

Solution steps

Find the critical points:

(2, 0), (0, 0), (1, 1), (1, - 1)

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

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

Check the sign of at each critical point

Check critical point

(2, 0) :

Minimum

Check critical point

(0, 0) :

Maximum

Check critical point

(1, 1) :

Saddle

Check critical point

(1, - 1) :

Saddle

Minimum (2, 0), Maximum (0, 0), Saddle (1, 1), Saddle (1, - 1)

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

40000 x + 30000 y - 8 x^{2} - 15 y^{2} - 10 x y

f (x) = \frac{- x ^{2}}{100} + x

f (x, y) = x^{2} y + y^{2} + 8 x

f (x) = x (15 - 48 + 2 x) (\frac{48}{2} - x)

What is the extreme F(x,y)=x^3+3xy^2-3y^2-3x^2+11 ?
The extreme F(x,y)=x^3+3xy^2-3y^2-3x^2+11 is Minimum(2,0),Maximum(0,0),Saddle(1,1),Saddle(1,-1)