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

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

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

extreme points

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

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = e^{y} (y^{2} + 4 y - x^{2} + 2)

D (x, y) = - 2 e^{2 y} x^{2} - 4 e^{2 y} - 2 e^{2 y} y^{2} - 8 e^{2 y} y

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(0, - 2) :

Maximum

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

y = x^{3}

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

\frac{2 x ^{2}}{x ^{2} - 25}

f (x) = \frac{x}{x ^{2} + 81}

F (I, J) = (25 I + 12 J) N

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