What is the extreme f(x,y)=(xy)/(1+e^x) ?

The extreme f(x,y)=(xy)/(1+e^x) is Saddle(0,0)

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extreme f(x,y)=(xy)/(1+e^x)

extreme points

f (x, y) = \frac{x y}{1 + e ^{x}}

Saddle (0, 0)

Solution steps

Find the critical points:

(0, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 0

D (x, y) = - \frac{( - e ^{x} x + e ^{x} + 1 ) ^{2}}{( e ^{x} + 1 ) ^{4}}

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Saddle (0, 0)

f (x, y) = 5 + 9 - x^{2} - y^{2}

f (x, y) = 2 x + y + \frac{2}{x} + \frac{4}{y}

f (x, y) = - x - y + 2

f (x, y) = 18 x y - 3 x^{2} y - 2 x y^{2}

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

What is the extreme f(x,y)=(xy)/(1+e^x) ?
The extreme f(x,y)=(xy)/(1+e^x) is Saddle(0,0)