What is the extreme f(x,y)=4xy-x^2y-xy^2 ?

The extreme f(x,y)=4xy-x^2y-xy^2 is Saddle(0,4),Maximum(4/3 , 4/3)

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extreme f(x,y)=4xy-x^2y-xy^2

extreme points

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

Saddle (0, 4), Maximum (\frac{4}{3}, \frac{4}{3})

Solution steps

Find the critical points:

(0, 4), (\frac{4}{3}, \frac{4}{3})

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

D (x, y) = - 4 x y + 16 x - 4 x^{2} + 16 y - 4 y^{2} - 16

Check the sign of at each critical point

Check critical point

(0, 4) :

Saddle

Check critical point

(\frac{4}{3}, \frac{4}{3}) :

Maximum

Saddle (0, 4), Maximum (\frac{4}{3}, \frac{4}{3})

What is the extreme f(x,y)=4xy-x^2y-xy^2 ?
The extreme f(x,y)=4xy-x^2y-xy^2 is Saddle(0,4),Maximum(4/3 , 4/3)