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Homework Help: To prove that these two functions meet only once

  1. Mar 11, 2010 #1
    1. The problem statement, all variables and given/known data

    How would you go about proving that the functions

    [tex] F(x)=\frac{2}{\pi}\arctan\Big(\frac{x}{a}\Big), x \geq 0[/tex]

    with [tex]a > 0[/tex]

    and

    [tex] G(x)=1-\exp(-\lambda x),x\geq0[/tex]

    with [tex]\lambda>0[/tex]


    meet only at one point for some [tex] x > 0 [/tex]

    3. The attempt at a solution

    At [tex]x=0[/tex], F and G takes the values 0s and both of the functions tend to 1 when x gets really large I think they must meet at least once. But still not sure about finding the actual point(s) since i do not think i can solve the equation
    [tex]F=G[/tex] analytically.
     
    Last edited: Mar 11, 2010
  2. jcsd
  3. Mar 11, 2010 #2
    as x becomes large, the two function tend to 1.
    And clearly they are increasing functions.

    so what do you suggest?
     
  4. Mar 11, 2010 #3
    I don't believe the problem is asking you to find x, rather to show it exists and is unique.
     
  5. Mar 11, 2010 #4
    Yeah showing the uniqueness of that particular point is enough.
     
    Last edited: Mar 11, 2010
  6. Mar 11, 2010 #5
    The fact that your functions have the same limit does not mean they ever meet. Take for instance 1/x and 1/(x+1), for x>0. Clearly they both tend to zero, and clearly they are never equal.

    I don't see right away how to do existence or uniqueness. By continuity if you can find a point where F<G then you have existence, and then maybe you could look at the derivative of F-G to show uniqueness. Or maybe there's something prettier. What topics has your classed covered recently?
     
  7. Mar 11, 2010 #6
    This question poped up somewhere in my research.
    I can use any results in calculus and analysis.

    Note I have made some correction to the F function.
     
    Last edited: Mar 12, 2010
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