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)

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