What is the extreme f(x,y)=7(x-y)e^{-x^2-y^2} ?

The extreme f(x,y)=7(x-y)e^{-x^2-y^2} is Minimum(-1/2 , 1/2),Maximum(1/2 ,-1/2)

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extreme f(x,y)=7(x-y)e^{-x^2-y^2}

extreme points

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

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 7 (- 4 e^{- x^{2} - y^{2}} y^{3} + 4 e^{- x^{2} - y^{2}} x y^{2} + 6 e^{- x^{2} - y^{2}} y - 2 e^{- x^{2} - y^{2}} x)

D (x, y) = 784 e^{- 2 x^{2} - 2 y^{2}} x^{4} y^{2} - 784 e^{2 (- x^{2} - y^{2})} x^{4} y^{2} - 392 e^{- 2 x^{2} - 2 y^{2}} x^{4} - 1568 e^{- 2 x^{2} - 2 y^{2}} x^{3} y^{3} + 1568 e^{2 (- x^{2} - y^{2})} x^{3} y^{3} + 784 e^{- 2 x^{2} - 2 y^{2}} x^{3} y + 784 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{4} - 784 e^{2 (- x^{2} - y^{2})} x^{2} y^{4} - 784 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{2} + 588 e^{- 2 x^{2} - 2 y^{2}} x^{2} - 196 e^{2 (- x^{2} - y^{2})} x^{2} + 784 e^{- 2 x^{2} - 2 y^{2}} x y^{3} - 1568 e^{- 2 x^{2} - 2 y^{2}} x y - 392 e^{- 2 x^{2} - 2 y^{2}} y^{4} + 588 e^{- 2 x^{2} - 2 y^{2}} y^{2} - 196 e^{2 (- x^{2} - y^{2})} y^{2}

Check the sign of at each critical point

Check critical point

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

Minimum

Check critical point

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

Maximum

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

y = x^{2} - 8 x

\frac{e ^{x}}{x}

f (x) = - x^{2} - y^{2} + 10 x + 10 y

2 cos (x) + cos^{2} (x)

What is the extreme f(x,y)=7(x-y)e^{-x^2-y^2} ?
The extreme f(x,y)=7(x-y)e^{-x^2-y^2} is Minimum(-1/2 , 1/2),Maximum(1/2 ,-1/2)