Let $f:[0,4\pi ]\to [0,\pi ]$ be defined by $f\left(x\right)$ $={\cos }^{-1}(\cos x)$ The number of points $x\in [0,4\pi ]$ satisfying the equation $f\left(x\right)=\frac{10-x}{10}$ is

We have

$\begin{array}{l}f\left(x\right)={\cos }^{-1}(\cos x)\\ =\left\{\begin{array}{ccc}{\cos }^{-1}(\cos x)& \text{ if }& 0\le x\le \pi \\ {\cos }^{-1}(\cos (2\pi -x\left)\right)& \text{ if }& \pi \le x\le 2\pi \\ {\cos }^{-1}(\cos (x-2\pi \left)\right)& \text{ if }& 2\pi \le x\le 3\pi \\ {\cos }^{-1}(\cos (4\pi -x\left)\right)& \text{ if }& 3\pi \le x\le 4\pi \end{array}\right.\\ =\left\{\begin{array}{cl}x& \text{ if }0\le x\le \pi \\ 2\pi -x& \text{ if }\pi \le x\le 2\pi \\ x-2\pi & \text{ if }2\pi \le x\le 3\pi \\ 4\pi -x& \text{ if }3\pi \le x\le 4\pi \end{array}\right.\end{array}$

Graph of $y=f\left(x\right)$ and $y=\frac{10-x}{10}=1-\frac{x}{10}=g\left(x\right)$ are shown in the following figure.

The above figure shows that the equations $f\left(x\right)=\frac{10-x}{x}$ has 3 solution.