6

for a final exam review for Real Analysis, the following problem is asked:

Let $f(x)$ be a differentiable function over $[0,1]$. Suppose that $f(0) = f(1) = 0$. Prove that $f'(x) - 2f(x)$ must have a zero inside of $(0,1)$.

My initial idea was to use Rolle's Theorem, since both endpoints are equal to 0 in order to show that $f'(x)$ must have a zero, but I'm not sure how to go from there. I've also tried to use the Cauchy Mean-Value Theorem by letting $g(x)$ be a function with $g'(x) = f'(x) - 2f(x)$, but I wasn't quite sure how to use that to prove the statement. We can't use any integration as the last chapter we've done is on differentiation. If anyone could point me in the right direction I'd appreciate it.

EnCt272cf2928267cbbc
  • 61
  • 11
    Whenever you see $f'-2f$ think $e^{-2x}$. Let $\phi(x)= e^{-2x} f(x)$. How would Rolle's theorem help here? – copper.hat Dec 18 '17 at 22:06
  • @copper.hat, that method is more "differential equations style" though, and is a great perspective, but would it be more likely that another technique is desired in the given context of what is probably an undergraduate introductory analysis course? I'm just asking because I'm curious myself. Of course a student in this course should have had ODEs probably. – jdods Dec 18 '17 at 22:15
  • @copper.hat Thanks for the suggestion. I see how this could lead to a solution, but based off of the nature of the course and the previous exams I'm not entirely sure if we're allowed to use a proof of this kind, and instead have to work out proofs derived from general theorems that we've learned in the course. I'll work with this for now and see if I can find anything out from it relating to the theorems that we've used. Thanks again! – EnCt272cf2928267cbbc Dec 18 '17 at 22:19
  • I don't know how else I would approach this with elementary tools. – copper.hat Dec 18 '17 at 22:27
  • Are you sure, that $f$ is just differentiable and not continuously differentiable? If $f$ is just differentiable, then $f'$ can be discontinuous and I think the statement will become wrong. – Mundron Schmidt Dec 18 '17 at 22:41
  • 2
    @copper.hat Your proposed solution is completely elementary. I think you should post an answer, and if there are howls of protest, we can look for something else. – zhw. Dec 18 '17 at 22:44
  • @MundronSchmidt did you read the post of copper.hat? – zhw. Dec 18 '17 at 22:45
  • @zhw.: I am curious if there is another elementary way :-). – copper.hat Dec 18 '17 at 22:58
  • The $2$ here is a red herring. Almost any function would do. – Hans Dec 18 '17 at 23:46
  • @Hans can you expand on this a bit? – EnCt272cf2928267cbbc Dec 19 '17 at 01:42
  • Given an arbitrary integrable function $g$, apply Rolle's theorem to $fe^{-\int_0^x g}$. – Hans Dec 19 '17 at 04:03
  • Your initial idea is correct. And all question of this kind is basically solve a simple ODE. This is surely the natural and elementary thought. – xbh Dec 19 '17 at 05:07

0 Answers0