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Popular >

extreme 6xye^{-x^2-y^2}

Solution

extreme points

6 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}} = 6 x (4 e^{- x^{2} - y^{2}} y^{3} - 6 e^{- x^{2} - y^{2}} y)

D (x, y) = - 288 e^{- 2 x^{2} - 2 y^{2}} x^{4} y^{2} - 144 e^{- 2 x^{2} - 2 y^{2}} x^{4} - 288 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{4} + 720 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{2} + 144 e^{- 2 x^{2} - 2 y^{2}} x^{2} - 144 e^{- 2 x^{2} - 2 y^{2}} y^{4} + 144 e^{- 2 x^{2} - 2 y^{2}} y^{2} - 36 e^{- 2 x^{2} - 2 y^{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})

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Frequently Asked Questions (FAQ)

What is the extreme 6xye^{-x^2-y^2} ?
The extreme 6xye^{-x^2-y^2} is