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), Maximum (1, 0), Maximum (- 1, 0), Minimum (0, 1), Minimum (0, - 1), Saddle (0, 0), Maximum (1, 0), Maximum (- 1, 0), Saddle (0, 0), Maximum (1, 0), Maximum (- 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 (- 2 e^{- x^{2} - y^{2}} x^{2} y^{2} + 2 e^{- x^{2} - y^{2}} y^{4} - 5 e^{- x^{2} - y^{2}} y^{2} + e^{- x^{2} - y^{2}} x^{2} + e^{- x^{2} - y^{2}})

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(1, 0) :

Maximum

Check critical point

(- 1, 0) :

Maximum

Check critical point

(0, 1) :

Minimum

Check critical point

(0, - 1) :

Minimum

Check critical point

(0, 0) :

Saddle

Check critical point

(1, 0) :

Maximum

Check critical point

(- 1, 0) :

Maximum

Check critical point

(0, 0) :

Saddle

Check critical point

(1, 0) :

Maximum

Check critical point

(- 1, 0) :

Maximum

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

Popular Examples

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f (x) = - 45 x^{2} + 400 x

extreme f(x)=3x^2-2x+5 extreme points

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

extreme f(x)=((2x^2+4x))/((2+x^2))extreme points

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

extreme f(x)=x^3+8x+12 extreme points

f (x) = x^{3} + 8 x + 12

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