What is the FUNCTION_MANY#extreme f(x,y)=6xy-2x^2y-3xy^2 ?

The FUNCTION_MANY#extreme f(x,y)=6xy-2x^2y-3xy^2 is Saddle(0,2),Maximum(1, 2/3)

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FUNCTION_MANY#extreme f(x,y)=6xy-2x^2y-3xy^2

extreme points

f (x, y) = 6 x y - 2 x^{2} y - 3 x y^{2}

Saddle (0, 2), Maximum (1, \frac{2}{3})

Solution steps

Find the critical points:

(0, 2), (1, \frac{2}{3})

\frac{\partial ^{2} f}{\partial x ^{2}} = - 6 x

D (x, y) = - 24 x y + 48 x - 16 x^{2} + 72 y - 36 y^{2} - 36

Check the sign of at each critical point

Check critical point

(0, 2) :

Saddle

Check critical point

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

Maximum

Saddle (0, 2), Maximum (1, \frac{2}{3})

What is the FUNCTION_MANY#extreme f(x,y)=6xy-2x^2y-3xy^2 ?
The FUNCTION_MANY#extreme f(x,y)=6xy-2x^2y-3xy^2 is Saddle(0,2),Maximum(1, 2/3)