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Intermideate value therem question

  1. Sep 11, 2011 #1
    prove that for c<0

    there is only one solution to

    [tex]xe^{\frac{1}{x}}=c[/tex]



    ??



    for x=1 we have f(1)>0



    the limit as x->-infinity is -infinity



    what to do?

    ?
     
  2. jcsd
  3. Sep 11, 2011 #2

    LCKurtz

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    Begin by drawing a graph to get an idea. In particular, what if x is less than 0 but close to 0?
     
  4. Sep 11, 2011 #3
    close to zero from minus
    its minus infinity

    when x goes to minus infinity its 0

    how it helps me?
     
  5. Sep 11, 2011 #4
    close to sero is 0
     
  6. Sep 11, 2011 #5

    LCKurtz

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    If c < 0 and you know your function approaches 0 as x → 0- and approaches -∞ a x → -∞, can you conclude your function = c for some x?
     
  7. Sep 12, 2011 #6
    i need to show that there is x1 f(x1)<c
    f(x2)>c


    from the limit when x goes to sero we get zero -e<f(x)<e
    from the limit when x goes minus infinity f(x)<-N

    what e to chhose?
    what N to choose?
     
  8. Sep 12, 2011 #7
    LCKurtz provided great hint, I'll try to help as well.

    To prove that the solution is single:
    If c<0 what can you conclude about the existence of the solution (to your function) in [tex][0,\infty)[/tex].
    In addition what can you tell about yours function behavior in [tex](-\infty,0)[/tex], how this helps you?

    To prove solution existence:
    See LKurtzs hint.
     
    Last edited: Sep 12, 2011
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