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

The extreme f(x,y)=4xy^2-2x^2y-x is Saddle(0, 1/2),Saddle(0,-1/2)

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

extreme points

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

Saddle (0, \frac{1}{2}), Saddle (0, - \frac{1}{2})

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 8 x

D (x, y) = - 16 x^{2} + 32 x y - 64 y^{2}

Check the sign of at each critical point

Check critical point

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

Saddle

Check critical point

(0, - \frac{1}{2}) :

Saddle

Saddle (0, \frac{1}{2}), Saddle (0, - \frac{1}{2})

f (x, y) = 14 x^{2} - 2 x^{3} + 4 x y

f (x) = \frac{1}{3} x^{3} - 5 x^{2} + 21 x - 16

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

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

f (x, y) = x^{3} + y^{2} - 6 x^{2} + y - 1

What is the extreme f(x,y)=4xy^2-2x^2y-x ?
The extreme f(x,y)=4xy^2-2x^2y-x is Saddle(0, 1/2),Saddle(0,-1/2)