Find and classify the stationary points

I have the following problem:

Find and classify the stationary points of:

f_{(x,y)}=ln(2x+y+2)-2x-y

First I calculate the partial derivatives and set them 0.

fx_{(x,y)}=2/(2x+y+2) - 2

fy_{(x,y)}=1/(y+2(x+1)) -1

Than I have the following system of linear equations:

2/(2x+y+2) - 2=0

1/(y+2(x+1)) -1=0

as result I get x=-(y+1)/2 and y=(-2x+1)

I can find no points, for which this holds... Did I do some mistake in my calculations or can I say that for this function I have no stationary points?

Re: Find and classify the stationary points

Quote:

Originally Posted by

**infernalmich** I have the following problem:

Find and classify the stationary points of:

f_{(x,y)}=ln(2x+y+2)-2x-y

First I calculate the partial derivatives and set them 0.

fx_{(x,y)}=2/(2x+y+2) - 2

fy_{(x,y)}=1/(y+2(x+1)) -1

Than I have the following system of linear equations:

2/(2x+y+2) - 2=0

which is the same as 2(x+ 1)+ y= 1

which is the same as 2(x+1)+ y= 1 again.

Quote:

as result I get x=-(y+1)/2 and y=(-2x+1)

This last equation is incorrect. It should be either y= -(2x+1) or y= -2x- 1.

Quote:

I can find no points, for which this holds... Did I do some mistake in my calculations or can I say that for this function I have no stationary points?

On the contrary, the two equations are equivalent so that **every** point on the line y= -2x- 1 is a stationary point.