What is the extreme f(x)=(x^2-4y^2)e^{1-x^2-y^2} ?

The extreme f(x)=(x^2-4y^2)e^{1-x^2-y^2} is

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Popular >

extreme f(x)=(x^2-4y^2)e^{1-x^2-y^2}

Solution

extreme points

f (x) = (x^{2} - 4 y^{2}) e^{1 - 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 (8 e^{- x^{2} - y^{2} + 1} y^{4} - 2 e^{- x^{2} - y^{2} + 1} x^{2} y^{2} - 20 e^{- x^{2} - y^{2} + 1} y^{2} + e^{- x^{2} - y^{2} + 1} x^{2} + 4 e^{- x^{2} - y^{2} + 1})

D (x, y) = - 4 (2 e^{- x^{2} - y^{2} + 1} x^{4} - 8 e^{- x^{2} - y^{2} + 1} x^{2} y^{2} - 5 e^{- x^{2} - y^{2} + 1} x^{2} + 4 e^{- x^{2} - y^{2} + 1} y^{2} + e^{- x^{2} - y^{2} + 1}) (8 e^{- x^{2} - y^{2} + 1} y^{4} - 2 e^{- x^{2} - y^{2} + 1} x^{2} y^{2} - 20 e^{- x^{2} - y^{2} + 1} y^{2} + e^{- x^{2} - y^{2} + 1} x^{2} + 4 e^{- x^{2} - y^{2} + 1}) - 4 x^{2} (- 8 e^{- x^{2} - y^{2} + 1} y^{3} + 2 e^{- x^{2} - y^{2} + 1} x^{2} y + 6 e^{- x^{2} - y^{2} + 1} y)^{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

extreme f(x,y)=x^3-3x+y^2-y^4 extreme points

f (x, y) = x^{3} - 3 x + y^{2} - y^{4}

extreme f(x)=(x-10)^{1/3}extreme points

f (x) = (x - 10)^{\frac{1}{3}}

extreme y=(2x-1)^2 extreme points

y = (2 x - 1)^{2}

extreme f(x)=3x+(147)/x extreme points

f (x) = 3 x + \frac{147}{x}

extreme f(x,y)=x^{(2)}+y^{(2)}+x^{(2)}y+4 extreme points

f (x, y) = x^{(2)} + y^{(2)} + x^{(2)} y + 4

Frequently Asked Questions (FAQ)

What is the extreme f(x)=(x^2-4y^2)e^{1-x^2-y^2} ?
The extreme f(x)=(x^2-4y^2)e^{1-x^2-y^2} is