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

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

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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}

Saddle (0, 1), Maximum (\frac{1}{3}, \frac{1}{3})

Solution steps

Find the critical points:

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

\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

(0, 1) :

Saddle

Check critical point

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

Maximum

Saddle (0, 1), Maximum (\frac{1}{3}, \frac{1}{3})

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

f (x, y) = x^{2} - 12 x y + 2 y^{4}

f (x, y) = - x^{3} + y^{2} + 3 x - 4 y - 1

f (x, y) = x^{3} + y^{3} - 9 x y

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

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