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

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

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

extreme points

f (x y) = 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}} = 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})

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

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

2 x^{2} + 4 x + 1

P (r, y) = (r + 2 1 + m^{2}) y

f (x) = - x^{2} + 6 x - 4