What is the extreme f(x,y)=21xye^{-x^2-y^2} ?

The extreme f(x,y)=21xye^{-x^2-y^2} is

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extreme f(x,y)=21xye^{-x^2-y^2}

extreme points

f (x, y) = 21 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}} = 21 x (4 e^{- x^{2} - y^{2}} y^{3} - 6 e^{- x^{2} - y^{2}} y)

D (x, y) = - 3528 e^{- 2 x^{2} - 2 y^{2}} x^{4} y^{2} - 1764 e^{- 2 x^{2} - 2 y^{2}} x^{4} - 3528 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{4} + 8820 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{2} + 1764 e^{- 2 x^{2} - 2 y^{2}} x^{2} - 1764 e^{- 2 x^{2} - 2 y^{2}} y^{4} + 1764 e^{- 2 x^{2} - 2 y^{2}} y^{2} - 441 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})

2 x^{3} - 3 x^{2} - x + 9

f (x) = (x - 6) e^{- 6 x}

2 t^{10} (4 - t^{2})^{5}

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

f (y) = x^{3} + y^{3} - 27 x y

What is the extreme f(x,y)=21xye^{-x^2-y^2} ?
The extreme f(x,y)=21xye^{-x^2-y^2} is