Find the number of real solutions to $\sin(4x)=\frac{x}{100}$ for $x \in (0,2\pi)$.
This was a question for a subject that only assumed basic high school knowledge, so the "right" way to solve it was by making the graphs for $\sin(4x)$ and $\frac{x}{100}$ and counting how many intersection points there are, which are $7$. However my class has discussed how to solve this question non-visually, we tried things like finding critical points, using the intermediate value theorem, using $\sin(2x)=2\sin(x)\cos(x)$, but we ended up nowhere. My question is, how can we solve this kind of problem non-visually, in a "college math" way?
Best Answer
A general approach to the types of problem where you want to find the number of solutions to some
$$f(x)=g(x)$$
is by instead defining $h(x)=f(x)-g(x)$ and then look for the number of solutions to
$$h(x)=0.$$
This is usually a much more approachable problem when you can't solve it directly. If you then also have continuous differentiability, what you can do is look for extremals by solving
$$h'(x)=0$$
as there can be at most one solution between these extremals as your function is continuous. So what you do is you find your extremals, let's call them $x_1,\dots,x_n$, where we order them by value, and look at the signs of these. If you have that $h(x_i)$ and $h(x_{i+1})$ have opposite signs, then you know that you have a solution between them, and if they have the same sign, you know that there is not a solution between them. So this for all of them and you will have found most possible solutions to the equation. The final thing to do is to investigate what happens before the first one and after the last one, and for that you could simply investigate the sign of the derivative there (it will not change as we are assuming continuous differentiability), and the sign of the function evaluated at the first and last extremals.
Now this was how you can take a general approach, and so for your problem you would define
$$h(x)=\sin(4x)-\frac{x}{100},$$
and so have that
$$h'(x)=4\cos(4x)-\frac{1}{100}.$$
So now you find the extremals of $h$ by solving (or at least investigating)
$$4\cos(4x)-\frac{1}{100}=0.$$
Now try to extract the relevant information from this.
Hopefully this helps!