A question based on property of a function satisfying $f(1/n) =0$ for every $n \in\mathbb{N} $

Question

I am trying quiz questions of senior year and was unable to solve this particular question.

It's image:

Unfortunately, I couldn't think which result in analysis I can use. I am totally confused and would really appreciate if someone tells what should be done in such kind of problems.

Kindly help.

Answers :

A, B, C

See this post for the arguments. – Dietrich Burde Jul 08 '20 at 19:10 — Dietrich Burde, Jul 08 '20 at 19:10

Rushabh Mehta · Accepted Answer · 2020-07-08T19:15:08.770

1

It's obvious that $D$ is not true since there are infinitely many roots, which a non-zero polynomial can't have.

$A$ is true by continuity, as $\frac1n$ approachs $0$.

By the Mean Value Theorem, for each interval $\left[\frac1{n+1},\frac1n\right]$, there is a point in the interval with derivative $0$. As these points approach $0$, by continuity, $B$ is true.

The same argument using MVT twice shows that $C$ is true.

edited Jul 08 '20 at 19:15

answered Jul 08 '20 at 19:09

Rushabh Mehta

13,663

You don't need MVT.... – Angina Seng Jul 08 '20 at 19:25
@AnginaSeng: I think we need MVT for $f''(0)=0$. The first derivative $f' (0)=0$ can be handled by definition. – Paramanand Singh Jul 09 '20 at 04:47

A question based on property of a function satisfying $f(1/n) =0$ for every $n \in\mathbb{N} $

1 Answers1