What is the extreme f(x)=9xye^{-x^2-y^2} ?

The extreme f(x)=9xye^{-x^2-y^2} is

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extreme f(x)=9xye^{-x^2-y^2}

extreme points

f (x) = 9 x y e^{- x^{2} - y^{2}}

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}} = 9 x (4 e^{- x^{2} - y^{2}} y^{3} - 6 e^{- x^{2} - y^{2}} y)

D (x, y) = - 648 e^{- 2 x^{2} - 2 y^{2}} x^{4} y^{2} - 324 e^{- 2 x^{2} - 2 y^{2}} x^{4} - 648 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{4} + 1620 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{2} + 324 e^{- 2 x^{2} - 2 y^{2}} x^{2} - 324 e^{- 2 x^{2} - 2 y^{2}} y^{4} + 324 e^{- 2 x^{2} - 2 y^{2}} y^{2} - 81 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})

f (x) = x^{3} - 12 x - 1

f (x, y) = 9 - x^{2} - 9 y^{2}

6 sin (x)

2 t^{3} - 24 t^{2} + 70

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