2

Given a function $f(x)$, such that for some $ b > a $, the following holds: $f(b) - f(a) = 0$

Is the following true?

For any $s < (b - a)$, there exist $c$ and $d$ (where $d > c$ and $d - c = s$) such that: $f(d) - f(c) = 0$

In my claim I'm assuming that f(x) is continuous function and c or d or (c and d both) can be outside a-b interval

informal claim: If a stock’s total net growth over a 25-year period is zero, then there must be at least one interval of length 20 years (and 15 years, 10 years, 5 years, etc.) over which net growth is also zero.

Konstantin
  • 21
  • What have you tried? – Duong Ngo Jan 11 '25 at 10:30
  • 2
    I tried LLM (chatGpt), but different models gives different answers - that is why I'm here :-) – Konstantin Jan 11 '25 at 10:48
  • Do $c$ and $d$ have to be in the interval $[a, b]$? In other words, is it required that $a<c<d<b$? – Mark H Jan 11 '25 at 11:34
  • no - no such requirement - c or d or both c and d can be outside a-b interval – Konstantin Jan 11 '25 at 12:32
  • Does $f: \mathbb{R} \to \mathbb{R}$ have any constraints? – Duong Ngo Jan 11 '25 at 12:58
  • The claim is inspired by this: If a stock’s total net growth over a 25-year period is zero, then there must be at least one interval of length 20 years (and 15 years, 10 years, 5 years, etc.) over which net growth is also zero. So the only constraint I can come up right now - is f(x) is continuous function – Konstantin Jan 11 '25 at 13:05
  • 1
    This is known as the universal chord theorem. It is true for $5$ years but not for the other periods. It is also true for periods of $\frac {25}n$ where $n$ is any natural greater than $1$. Another question on it is https://math.stackexchange.com/questions/612462/intuition-for-the-universal-chord-theorem. Another is https://math.stackexchange.com/questions/113346/prove-that-if-two-miles-are-run-in-759-then-one-mile-must-be-run-under-400 – Ross Millikan Jan 11 '25 at 14:37
  • @RossMillikan, thank you for referencing the universal chord theorem. Could you tell me if it's easy to find a continuous f(x) that contradicts my claim? if so how to find the f(x). – Konstantin Jan 11 '25 at 18:44
  • 1
    There is an answer to the question I marked as a duplicate by Martin Sleziak that has a sketch of one and some links. – Ross Millikan Jan 11 '25 at 20:36
  • 1
    Adapting the formula at the end of this answer to the interval $[0,25]$ (instead of $[0,1]$), for $20$ years we get $f(x)=\sin^2\left(\frac\pi{20}x\right)-\frac1{50}x.$ – David K Jan 12 '25 at 00:38
  • Just in case, I tried to find 20 year chord on [-25, 50] years interval for the sin^2 function mentioned - and I could not find it. @DavidK thank you! – Konstantin Jan 12 '25 at 16:42

0 Answers0