1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]


    [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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook