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

The extreme f(x,y)=(x^2+y^2)e^{y^2-x^2} is

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

extreme points

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

Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0), Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0), Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0)

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 (2 e^{(x + y) (- x + y)} y^{4} + 2 e^{(x + y) (- x + y)} x^{2} y^{2} + 5 e^{(x + y) (- x + y)} y^{2} + e^{(x + y) (- x + y)} x^{2} + e^{(x + y) (- x + y)})

D (x, y) = 4 (2 e^{(x + y) (- x + y)} x^{4} + 2 e^{(x + y) (- x + y)} x^{2} y^{2} - 5 e^{(x + y) (- x + y)} x^{2} - e^{(x + y) (- x + y)} y^{2} + e^{(x + y) (- x + y)}) (2 e^{(x + y) (- x + y)} y^{4} + 2 e^{(x + y) (- x + y)} x^{2} y^{2} + 5 e^{(x + y) (- x + y)} y^{2} + e^{(x + y) (- x + y)} x^{2} + e^{(x + y) (- x + y)}) - 4 x^{2} (- 2 e^{(x + y) (- x + y)} y^{3} - 2 e^{(x + y) (- x + y)} x^{2} y)^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(1, 0) :

Saddle

Check critical point

(- 1, 0) :

Saddle

Check critical point

(0, 0) :

Minimum

Check critical point

(1, 0) :

Saddle

Check critical point

(- 1, 0) :

Saddle

Check critical point

(0, 0) :

Minimum

Check critical point

(1, 0) :

Saddle

Check critical point

(- 1, 0) :

Saddle

Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0), Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0), Minimum (0, 0), Saddle (1, 0), Saddle (- 1, 0)

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

f (x) = 3 x - 2 x^{2} - \frac{4 x ^{3}}{3}

f (x, y) = 2 x^{2} + 3 y^{2} - 7

f (x) = \frac{x ^{2} + 4}{x}

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

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