f(x) is defined for the interval [a, b] and continuous, where f(a) = f(b) = 5 and b - a = 24, show that there exist x such that f(x) = f(x + 12).

Question

I know that this question can be solved using Intermediate Value Theorem, but I don't know how to show that such a point exist between the interval. Ideally, it would be very helpful to show that between the interval there exists x for which f(x) = f(x+c) : 0 <= c <= 24.

I assume that $f$ is continuous, but you haven't actually stated that. — Robert Shore, May 19 '22 at 01:35

score 1 · Answer 1 · answered May 19 '22 at 01:36

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I will assume that $f$ is continuous. Let $g(x)=f(x+12)-f(x)$. Then, we have $g(b-12)=f(b)-f(b-12)$ and $g(a)=f(a+12)-f(a)$. As $b-12=a+12$, we have $f(a+12)=f(b-12)=c$. If $c=5$, then we are done; $x=a$ or $x=a+12=b-12$ is the $c$ we are looking for. If $c\neq 5$, then $g(a)g(b-12)=(c-5)(5-c)=-(c-5)^2<0$ and so we have either $g(a)>0, g(b-12)<0$ or $g(a)<0, g(b-12)>0$. Then, what can we say by IVT?

answered May 19 '22 at 01:36

Joshua Woo

1,183

That at some point, g(x) = 0? – mmx May 19 '22 at 02:40

f(x) is defined for the interval [a, b] and continuous, where f(a) = f(b) = 5 and b - a = 24, show that there exist x such that f(x) = f(x + 12).

1 Answers1