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

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

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

extreme points

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

Maximum (0, 0), Saddle (1, 2), Saddle (1, - 2)

Solution steps

Find the critical points:

(0, 0), (1, 2), (1, - 2)

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Maximum

Check critical point

(1, 2) :

Saddle

Check critical point

(1, - 2) :

Saddle

Maximum (0, 0), Saddle (1, 2), Saddle (1, - 2)

y = 5 x - x^{2} + 66

x^{2} - 8 x + 22

\frac{x ^{4} + 8}{x ^{2}}

f (x) = (x - 2)^{3} + 5

f (x, y) = 18 y^{2} + x^{2} - x^{2} y

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