How many solutions does $15\cos x = x$ have on $[0,5\pi]$?

Question

So I found that on $(1,2)$ and $(4,5)$ we have two roots by the Intermediate Value Theorem. Am I correct?

Edit 1: I miscalculated there can not be a root in $(4,5)$ when using the theorem.

And while from the graph there is indeed one root apparently but this question has options. It says that the number of roots are either $3$, $4$, $5$ or $6$.

Edit 2: So I messaged my teacher and they said that it was a typo, the interval is indeed $[0,5\pi]$.

Answer 1

There is only one root around $x=1.5$.

At the end of the interval, make a simple series expansion $$15\cos (x) - x=(15 \cos (5)-5)- (1+15 \sin (5))(x-5)+O\left((x-5)^2\right)$$Ignoring the higher order terms $$x \sim 5+\frac{5 (3 \cos (5)-1)}{1+15 \sin (5)}\approx 5.05567 \,\,\, >5$$

Edit

Taking into account the answer key, the problem is $$\Large[0,5\color{red}{\pi}]$$