What is the extreme ln(x+y)-ln(1+xy) ?

The extreme ln(x+y)-ln(1+xy) is Saddle(1,1),Saddle(-1,-1)

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extreme ln(x+y)-ln(1+xy)

extreme

Saddle (1, 1), Saddle (- 1, - 1)

Solution steps

Find the critical points:

(1, 1), (- 1, - 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{( - x ^{2} + 1 ) ( 2 x y + x ^{2} + 1 )}{( x + y ) ^{2} ( x y + 1 ) ^{2}}

D (x, y) = \frac{2 x ^{5} y ^{3} - 2 x ^{5} y + 4 x ^{4} y ^{4} - 6 x ^{4} y ^{2} - 2 x ^{4} + 2 x ^{3} y ^{5} - 8 x ^{3} y ^{3} - 10 x ^{3} y - 6 x ^{2} y ^{4} - 16 x ^{2} y ^{2} - 2 x ^{2} - 2 x y ^{5} - 10 x y ^{3} - 4 x y - 2 y ^{4} - 2 y ^{2}}{( x + y ) ^{4} ( 1 + x y ) ^{4}}

Check the sign of at each critical point

Check critical point

(1, 1) :

Saddle

Check critical point

(- 1, - 1) :

Saddle

Saddle (1, 1), Saddle (- 1, - 1)

What is the extreme ln(x+y)-ln(1+xy) ?
The extreme ln(x+y)-ln(1+xy) is Saddle(1,1),Saddle(-1,-1)