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

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

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

extreme points

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

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 1

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

x^{2} + 2 x + 5

f (x) = 420 t - 35 t

f (x) = x^{2} - 25

\frac{x ^{2}}{x + 2}

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

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