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

The extreme f(x,y)=x+xy^2-x^2y^3+y is Saddle(1,1),Saddle(-1,-1)

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extreme f(x,y)=x+xy^2-x^2y^3+y

extreme points

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

Saddle (1, 1), Saddle (- 1, - 1)

Solution steps

Find the critical points:

(1, 1), (- 1, - 1)

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

D (x, y) = - 24 x^{2} y^{4} + 20 x y^{3} - 4 y^{2}

Check the sign of at each critical point

Check critical point

(1, 1) :

Saddle

Check critical point

(- 1, - 1) :

Saddle

Saddle (1, 1), Saddle (- 1, - 1)

What is the extreme f(x,y)=x+xy^2-x^2y^3+y ?
The extreme f(x,y)=x+xy^2-x^2y^3+y is Saddle(1,1),Saddle(-1,-1)