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I have not found much online regarding this question. It's by no means my area of research, but is a questioning I came across when writing a program. If I have two exponential decay functions defined on the same interval $[0,1]$, how many intersection points can they have on this interval?

The two functions can be defined in the form (with $a,b,c,a',b',c' > 0$): \begin{align} f(x) = a + b\exp(-c x), && g(x) = a' + b'\exp(-c' x). \end{align} Assuming $f(x)$ and $g(x)$ are not equal, I seem to either find $0$, $1$ or $2$ intersection points. Would you have suggestions on how to approach this problem? Thank you!

Gonçalo
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  • Hint: How many points are needed to uniquely specify the three parameters $a, b, c$? – eyeballfrog Jul 12 '22 at 14:19
  • This follows directly from the theory of total positivity. If a signed measure has $k$ sign changes, its Laplace transform has at most $k$ positive roots. – kimchi lover Jul 02 '23 at 00:06

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