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

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

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

extreme points

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

Saddle (1, 0), Saddle (- 3, 0), Maximum (- 1, - 2)

Solution steps

Find the critical points:

(1, 0), (- 3, 0), (- 1, - 2)

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

D (x, y) = - 4 y - 4 x^{2} - 8 x - 4

Check the sign of at each critical point

Check critical point

(1, 0) :

Saddle

Check critical point

(- 3, 0) :

Saddle

Check critical point

(- 1, - 2) :

Maximum

Saddle (1, 0), Saddle (- 3, 0), Maximum (- 1, - 2)

f (x) = 5 sin (2 x) + 3

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

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

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

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

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