en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

extreme xye^{-x^2-y^2}

Solution

extreme points

x y e^{- x^{2} - y^{2}}

Solution

Saddle (0, 0), Maximum (- \frac{1}{2}, - \frac{1}{2}), Minimum (\frac{1}{2}, - \frac{1}{2}), Minimum (- \frac{1}{2}, \frac{1}{2}), Maximum (\frac{1}{2}, \frac{1}{2})

Solution steps

Find the critical points:

(0, 0), (- \frac{1}{2}, - \frac{1}{2}), (\frac{1}{2}, - \frac{1}{2}), (- \frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2})

\frac{\partial ^{2} f}{\partial x ^{2}} = x (4 e^{- x^{2} - y^{2}} y^{3} - 6 e^{- x^{2} - y^{2}} y)

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- \frac{1}{2}, - \frac{1}{2}) :

Maximum

Check critical point

(\frac{1}{2}, - \frac{1}{2}) :

Minimum

Check critical point

(- \frac{1}{2}, \frac{1}{2}) :

Minimum

Check critical point

(\frac{1}{2}, \frac{1}{2}) :

Maximum

Saddle (0, 0), Maximum (- \frac{1}{2}, - \frac{1}{2}), Minimum (\frac{1}{2}, - \frac{1}{2}), Minimum (- \frac{1}{2}, \frac{1}{2}), Maximum (\frac{1}{2}, \frac{1}{2})

Popular Examples

extreme f(x)=x^4-32x+4 extreme points

f (x) = x^{4} - 32 x + 4

extreme f(x,y)=12xy-x^2-3y^2 extreme points

f (x, y) = 12 x y - x^{2} - 3 y^{2}

extreme f(x,y)=4xy^2-2x^2-16y^2 extreme points

f (x, y) = 4 x y^{2} - 2 x^{2} - 16 y^{2}

extreme f(x)=(3x+1)/(1-2x)extreme points

f (x) = \frac{3 x + 1}{1 - 2 x}

extreme f(x,y)=16x+12y-2x^2-3y^2 extreme points

f (x, y) = 16 x + 12 y - 2 x^{2} - 3 y^{2}

Frequently Asked Questions (FAQ)

What is the extreme xye^{-x^2-y^2} ?
The extreme xye^{-x^2-y^2} is