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

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

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

extreme points

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

Minimum (- \frac{1}{3}, \frac{1}{3}), Saddle (0, 1)

Solution steps

Find the critical points:

(- \frac{1}{3}, \frac{1}{3}), (0, 1)

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

D (x, y) = 4 x y - 4 x - 4 x^{2} + 4 y - 4 y^{2} - 1

Check the sign of at each critical point

Check critical point

(- \frac{1}{3}, \frac{1}{3}) :

Minimum

Check critical point

(0, 1) :

Saddle

Minimum (- \frac{1}{3}, \frac{1}{3}), Saddle (0, 1)

f (x) = x^{3} + 6 x^{2} + 12 x + 2

f (x) = - 0.002 x^{2} + 3.2 x - 60

f (x) = x^{3} + 6 x^{2} + 12 x + 3

f (x) = 7 x^{4} - 28 x^{3} + 5

f (x) = x^{4} - 2 4^{2} + 180

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