What is the extreme f(x,y)=xy(1-x-y) ?

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

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extreme f(x,y)=xy(1-x-y)

extreme points

f (x, y) = x y (1 - x - y)

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^{2} + 4 x - 4 y^{2} + 4 y - 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})

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