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

The extreme f(x,y)=xln(x^2+y^2) is Saddle(0,1),Saddle(0,-1)

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extreme f(x,y)=xln(x^2+y^2)

extreme points

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

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

Solution steps

Find the critical points:

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

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

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

Check the sign of at each critical point

Check critical point

(0, 1) :

Saddle

Check critical point

(0, - 1) :

Saddle

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

f (x) = x^{\frac{12}{5}} - 6 x^{\frac{2}{5}} + 2

10 x - 5 x ln (x)

f (x) = x + 4 x^{- 1}

y = I n 9 - 2 x^{2}

f (x) = 2 x^{3} - 33 x^{2} + 144 x

What is the extreme f(x,y)=xln(x^2+y^2) ?
The extreme f(x,y)=xln(x^2+y^2) is Saddle(0,1),Saddle(0,-1)