What is the extreme g(x,y)=(x^2+y)/(x^2+y^2+1) ?

The extreme g(x,y)=(x^2+y)/(x^2+y^2+1) is Saddle(0,1),Minimum(0,-1)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

extreme g(x,y)=(x^2+y)/(x^2+y^2+1)

extreme

Saddle (0, 1), Minimum (0, - 1)

Solution steps

Find the critical points:

(0, 1), (0, - 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{2 ( x ^{4} + 3 x ^{2} y + x ^{2} - 3 x ^{2} y ^{2} + 3 y - y ^{3} )}{( x ^{2} + y ^{2} + 1 ) ^{3}}

D (x, y) = \frac{4 y ^{7} - 4 x ^{2} y ^{6} - 4 y ^{6} + 12 x ^{2} y ^{5} - 4 y ^{5} - 8 x ^{4} y ^{4} - 24 x ^{2} y ^{4} + 8 y ^{4} + 12 x ^{4} y ^{3} + 24 x ^{2} y ^{3} - 20 y ^{3} - 4 x ^{6} y ^{2} - 12 x ^{4} y ^{2} - 12 x ^{2} y ^{2} + 12 y ^{2} + 4 x ^{6} y + 28 x ^{4} y + 12 x ^{2} y - 12 y + 8 x ^{6} - 8 x ^{2}}{( x ^{2} + y ^{2} + 1 ) ^{6}}

Check the sign of at each critical point

Check critical point

(0, 1) :

Saddle

Check critical point

(0, - 1) :

Minimum

Saddle (0, 1), Minimum (0, - 1)

What is the extreme g(x,y)=(x^2+y)/(x^2+y^2+1) ?
The extreme g(x,y)=(x^2+y)/(x^2+y^2+1) is Saddle(0,1),Minimum(0,-1)