What is the extreme f(x,y)=ln(x^2+y^2+1) ?

The extreme f(x,y)=ln(x^2+y^2+1) is Minimum(0,0)

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extreme f(x,y)=ln(x^2+y^2+1)

extreme points

f (x, y) = ln (x^{2} + y^{2} + 1)

Minimum (0, 0)

Solution steps

Find the critical points:

(0, 0)

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Minimum (0, 0)

What is the extreme f(x,y)=ln(x^2+y^2+1) ?
The extreme f(x,y)=ln(x^2+y^2+1) is Minimum(0,0)