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

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

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

extreme points

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

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 x - 4

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Maximum

Check critical point

(2, 2) :

Saddle

Check critical point

(2, - 2) :

Saddle

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

f (x) = - x^{4} - 17 x^{3} - 106 x^{2} - 290 x - 293

f (x) = sin (4 x), - \frac{π}{6} \leq x \leq \frac{π}{4}

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

p (a, c) = a + 2 + c

f (x) = 40 e^{2 x} - 5 e^{x}

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