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To prove that these two functions meet only once

  • Thread starter willy0625
  • Start date
  • #1
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Homework Statement



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]

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:

Answers and Replies

  • #2
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Homework Statement



How would you go about proving that the functions

[tex] F(x)=\frac{1}{2}+\frac{1}{\pi}\arctan\Big(\frac{x}{a}\Big), x\in\mathbb{R}[/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]

The Attempt at a Solution



At [tex]x=0[/tex], F and G takes the values 0.5 a 0, respectively and both of the functions tend to 1 when x gets really large I think they must meet a 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.
as x becomes large, the two function tend to 1.
And clearly they are increasing functions.

so what do you suggest?
 
  • #3
236
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I don't believe the problem is asking you to find x, rather to show it exists and is unique.
 
  • #4
6
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Yeah showing the uniqueness of that particular point is enough.
 
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  • #5
236
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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?
 
  • #6
6
0
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:

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