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

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

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extreme f(x)=xln(y^2-x)

extreme points

f (x) = x ln (y^{2} - x)

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 ( - y ^{2} - x )}{( y ^{2} - x ) ^{2}}

D (x, y) = \frac{2 x ( - 2 y ^{2} + x ) ( - y ^{2} - x ) - 4 y ^{6}}{( y ^{2} - x ) ^{4}}

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^{4} + 2 x^{2}

f (x) = x^{3} - 12 x^{2}

f (x) = 2 x^{3} + 3 x^{2} - 336 x

y = x^{4} + 6 x^{3} + 12 x^{2} + 8 x + 10

f (x, y) = x^{2} - 2 x y + 3 y^{2}

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