What is the extreme f(x,y)=(xy(9-x-2y))/3 ?

The extreme f(x,y)=(xy(9-x-2y))/3 is Saddle(0, 9/2),Maximum(3, 3/2)

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extreme f(x,y)=(xy(9-x-2y))/3

extreme points

f (x, y) = \frac{x y ( 9 - x - 2 y )}{3}

Saddle (0, \frac{9}{2}), Maximum (3, \frac{3}{2})

Solution steps

Find the critical points:

(0, \frac{9}{2}), (3, \frac{3}{2})

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{4 x}{3}

D (x, y) = \frac{- 4 x ^{2} - 8 x y + 36 x - 16 y ^{2} + 72 y - 81}{9}

Check the sign of at each critical point

Check critical point

(0, \frac{9}{2}) :

Saddle

Check critical point

(3, \frac{3}{2}) :

Maximum

Saddle (0, \frac{9}{2}), Maximum (3, \frac{3}{2})

x^{3} + 3 x y^{2} - 3 y^{2} - 3 x^{2} + 11

0.001 x^{2} + 3.4 x - 90

f (x) = 12 x - 12 y - 2 x^{2} - 3 y^{2} - 25

f (x) = 3 x^{3} + 9 x^{2} - 72 x

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

What is the extreme f(x,y)=(xy(9-x-2y))/3 ?
The extreme f(x,y)=(xy(9-x-2y))/3 is Saddle(0, 9/2),Maximum(3, 3/2)