1. Interpret the function carefully

The notation $f(x)=\cos^{-1}(\cos x)$ means $f(x)=\arccos(\cos x),$ with principal value in $[0,\pi]$.

We must solve $\arccos(\cos x)=\frac{10-x}{10}, \qquad x\in[0,4\pi].$

Let $y=\frac{10-x}{10}=1-\frac{x}{10}.$ Since the left side lies in $[0,\pi]$, we must also have $0\le 1-\frac{x}{10}\le \pi.$ From this, $0\le 1-\frac{x}{10} \implies x\le 10.$ So any solution must satisfy $x\in[0,10].$ Because $4\pi\approx 12.57$, this reduces the search interval to $[0,10]$.

2. Write $\arccos(\cos x)$ piecewise on $[0,4\pi]$

The standard triangular-wave form is:

• for $x\in[0,\pi]$, $\arccos(\cos x)=x$
• for $x\in[\pi,2\pi]$, $\arccos(\cos x)=2\pi-x$
• for $x\in[2\pi,3\pi]$, $\arccos(\cos x)=x-2\pi$
• for $x\in[3\pi,4\pi]$, $\arccos(\cos x)=4\pi-x$

But since solutions must satisfy $x\le 10$, only the first three pieces matter, because $10<4\pi$ and also $10>3\pi$.

So solve piecewise on: $[0,\pi],\quad [\pi,2\pi],\quad [2\pi,3\pi],\quad [3\pi,10].$

3. Solve on $[0,\pi]$

Here, $f(x)=x.$ Equation becomes $x=1-\frac{x}{10}.$ So $x+\frac{x}{10}=1$ $\frac{11x}{10}=1$ $x=\frac{10}{11}.$ This lies in $[0,\pi]$, so it is a valid solution.

4. Solve on $[\pi,2\pi]$

Here, $f(x)=2\pi-x.$ Equation becomes $2\pi-x=1-\frac{x}{10}.$ Multiply by $10$: $20\pi-10x=10-x$ $20\pi-10=9x$ $x=\frac{20\pi-10}{9}.$ Now check whether this lies in $[\pi,2\pi]$.

Numerically, $x\approx \frac{62.83-10}{9}=\frac{52.83}{9}\approx 5.87,$ and since $\pi\approx 3.14,\qquad 2\pi\approx 6.28,$ this is valid.

So this gives a second solution.

5. Solve on $[2\pi,3\pi]$

Here, $f(x)=x-2\pi.$ Equation becomes $x-2\pi=1-\frac{x}{10}.$ So $x+\frac{x}{10}=1+2\pi$ $\frac{11x}{10}=1+2\pi$ $x=\frac{10(1+2\pi)}{11}.$ Numerically, $x\approx \frac{10(1+6.283)}{11}=\frac{72.83}{11}\approx 6.62.$ Now $2\pi\approx 6.28,\qquad 3\pi\approx 9.42,$ so this lies in $[2\pi,3\pi]$.

Hence this is a third solution.

6. Solve on $[3\pi,10]$

Here, $f(x)=4\pi-x.$ Equation becomes $4\pi-x=1-\frac{x}{10}.$ So $4\pi-1=\frac{9x}{10}$ $x=\frac{10(4\pi-1)}{9}.$ Numerically, $x\approx \frac{10(12.57-1)}{9}=\frac{115.7}{9}\approx 12.86.$ But this is not in $[3\pi,10]$ (in fact it exceeds $10$), so no valid solution from this branch.

7. Count the solutions

Valid solutions are: $x=\frac{10}{11},\qquad x=\frac{20\pi-10}{9},\qquad x=\frac{10(1+2\pi)}{11}.$ Thus, the number of points is $\boxed{3}.$

8. Comparison with stored answer

Stored correct answer = $3$.

Our derived answer is also $3$, so they agree.