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Math Help - Find and classify the stationary points

  1. #1
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    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?
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  2. #2
    MHF Contributor

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    Re: Find and classify the stationary points

    Quote Originally Posted by infernalmich View Post
    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.
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