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

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

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Popular >

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

Solution

extreme points

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

Solution

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 e^{- x^{2} - y^{2}} (2 y^{4} - 2 x^{2} y^{2} - 5 y^{2} + x^{2} + 1)

D (x, y) = 4 e^{- 2 x^{2} - 2 y^{2}} (- 2 x^{4} + 2 x^{2} y^{2} + 5 x^{2} - y^{2} - 1) (2 y^{4} - 2 x^{2} y^{2} - 5 y^{2} + x^{2} + 1) - 16 e^{2 (- x^{2} - y^{2})} x^{2} y^{2} (y + x)^{2} (y - x)^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- 1, 0) :

Minimum

Check critical point

(1, 0) :

Minimum

Check critical point

(0, - 1) :

Maximum

Check critical point

(0, 1) :

Maximum

Check critical point

(0, 0) :

Saddle

Check critical point

(- 1, 0) :

Minimum

Check critical point

(1, 0) :

Minimum

Check critical point

(0, 0) :

Saddle

Check critical point

(- 1, 0) :

Minimum

Check critical point

(1, 0) :

Minimum

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

Graph

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Frequently Asked Questions (FAQ)

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