Tessellate S.T.E.M.S - Mathematics - College - Set 1 - Problem 3

Calculus Level 3

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function such that $f$ vanishes at uncountably many points of $\mathbb{R}$ . Let $f^{(m)}(x)$ denote the derivative of $f(x)$ $m$ times.

Which of the following options are correct?

The subset $\left \{x \in \mathbb{R} : f^{(m)}(x)=0 \hspace{4pt} \forall \hspace{2pt} m \in \mathbb{N} \cup \{0\} \right \}$ is non-empty.
The subset $\left \{x \in \mathbb{R} : f^{(m)}(x)=0 \hspace{4pt} \forall \hspace{2pt} m \in \mathbb{N} \cup \{0\} \right \}$ is infinite.
The subset $\left \{x \in \mathbb{R} : f^{(m)}(x)=0 \hspace{4pt} \forall \hspace{2pt} m \in \mathbb{N} \cup \{0\} \right \}$ is infinite but not necessarily uncountable.

This problem is a part of Tessellate S.T.E.M.S.

Only 1 2 and 3 None of the above 1 and 2

2 solutions

Nilesh Khatri
Jan 10, 2019

Does the following function satisfy the given conditions? f(x)=x if x is rational f(x)=0 if x is irrational

It does not, because it's not infinitely differentiable.

Joe Mansley - 5 months, 2 weeks ago

Max Weinstein
Apr 18, 2018

This is a tentative solution. I'm not sure how rigorous this is.

First, what would the graph of $f(x)$ look like?

Well, it vanishes at uncountably many points, so does it cross the x-axis infinitely many times?

Sort of. As an example, $sin(x)$ crosses the x-axis infinitely many times. However, it only does so when $x=n\pi, n\in\mathbb{N}$ , so its zeroes are in one-to-one correspondence with $\mathbb{N}$ , meaning $sin(x)$ only vanishes a countable number of times. Actually, for any $g(x)$ which only crosses the x-axis at disjoint points, $g(x)$ only crosses the x-axis a countable number of times.

[note: This part I'm not quite sure how to prove. That's why this solution is tentative]

So, we must be talking about functions $f(x)$ which intersect the x-axis over the course of an interval $I$ (every interval contains an uncountable number of x-values).

However, every tangent line to a point in $I$ will have the equation $y=0$ ( $f(I)$ is a flat line), which means the graph of $f'(x)$ will also intersect the x-axis over $I$ . Additionally, because $f'(I)$ is a flat line, $f''(I)=0$ is, too, and this can be applied for any order of derivative, so we have:

x-values at which every order derivative is zero (statement 1 is true)
an uncountable number of these x-values (statement 2 is true, statement 3 is false)

Tessellate S.T.E.M.S - Mathematics - College - Set 1 - Problem 3

2 solutions

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