Thread: 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?

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

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

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

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.

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.