Strictly Increasing Function With Periodic First Derivative

Question

I was tasked with the following question:

Construct the functions F such that

1. F has domain R
2. F is differentiable
3. F is strictly increasing.
4. F′ is periodic with period 2π, and F′ is not constant.

This is what I've been able to come up with so far:

Let $a \in \mathbb{R}$ and assume $a \neq 0$. Let $F(x)$ be a function such that $F(x)=a\sin x + Cx$ s.t. $C > |a|$.

1. Showing that $F$ has domain $\mathbb{R}$:
   It can be shown that $\forall x \in \mathbb{R}$, $a \sin x$ and $C x$ exist, hence $\forall x \in \mathbb{R}, F(x)$ exists, which means $F$ has domain $\mathbb{R}$.

2. WTS $F$ is diffable.
   We find that $$F'(x)=a \cos x + C.$$
   It can be seen that $\forall x \in \mathbb{R}, a \cos x$ exists and $C$ exists. Hence $\forall x \in \mathbb{R}, F'(x)$ exists, which means $F$ is diffable.

3. Showing that $F$ is strictly increasing.
   1st. Assume $a>0$. We know $C>|a|\ge a$. Since $-1 \le \cos x$, $a \cos x + C \ge -a + C > -a + a = 0$, which means $F'(x)>0$.
   2nd. Assume $a<0$. We know $C>|a|\ge -a$. Since $\cos x \le 1$, $a \cos x + C > a + C > a - a = 0$, which means $F'(x)>0$.
   Hence $\forall a \neq 0, F'(x)>0 \ $for all x$ \in \mathbb{R} $. By MVT, this means $F$ is strictly increasing.

4. Prove that $F'$ is periodic with period $2\pi$ and $F'$ is not constant.
   Let's look at $$F'(x)=a \cos x + C.$$
   It can be shown that $F'(0)=a+C$ while $F'(\pi/2)=C$. Since $a \neq 0$, $F'(0) \neq F'(\pi/2)$, thus $F'$ is not constant. Note that $\cos x$ is periodic with period $2\pi$. Since $a \neq 0$ and $C \in \mathbb{R}$, thus by trigonometric properties, $F'(x)=a \cos x + C$ is also periodic.

My confusion: It's not necessary that $C > |a|$ (or $F'(x)>0$) ,because $C=|a|$ can work too: Think about $F(x)=\sin x + x$. It's still strictly increasing even though $C =|a|$ Thus there are points where $F'(x)=0$. However, I would not be able to prove it using the corollary that positive derivative implies strictly increasing and have to prove from the definition of increasing. I couldn't achieve this.

Answer 1

You seem to be trying to find all functions of a certain class that have the necessary properties. Why not just find one, as required. You're on the right track. Now ask yourself, "Would choosing $a = 1$ be viable? Because it'd sure make the algebra nicer!" Then you see that you need $C \ge |a|$, but the "=" case is a little touchy...so why not choose a value of $C$ that's solidly greater than $a$, and easy to work with arithmetically/algebraically.

Once you've chosen a specific function, you just have to show it's differentiable and periodic and increasing. ("Defined on the reals" really doesn't need proof -- sine and $x \to Cx$ are both defined on the reals, and you can just add them: the sum of two functions on $\mathbb R$ is again a function on $\mathbb R$.)

I hope this helps you focus your work.

Answer 2

I think I get it for Question 1: since F' is positive everywhere else (like on intervals (−π,π),(π,3π) ) , and since F is continuous on each of the intervals …,[−3π,−π],[−π,π],[π,3π],…, By the MVT F(x) is strictly increasing on each of the intervals …,[−3π,−π],[−π,π],[π,3π],… and hence on R.