Let `f:[0,4pi]->[0,pi]` be defined by `f(x)=cos^-1(cos x).` The number of points `x in[0,4pi]` 4satisfying the equation `f(x)=(10-x)/10` is

Explanation

To solve the problem, we need to find the number of points \( x \) in the interval \([0, 4\pi]\) that satisfy the equation:

\[ f(x) = \frac{10 - x}{10} \]

where \( f(x) = \cos^{-1}(\cos x) \).

### Step 1: Understand the function \( f(x) \)

The function \( f(x) = \cos^{-1}(\cos x) \) gives the principal value of the angle whose cosine is \( \cos x \). The range of \( f(x) \) is \([0, \pi]\) for \( x \) in the interval \([0, 4\pi]\).

### Step 2: Determine the behavior of \( f(x) \)

The function \( f(x) \) is periodic with a period of \( 2\pi \). Thus, we can analyze \( f(x) \) over one period, say \([0, 2\pi]\), and then extend our findings to \([0, 4\pi]\).

- For \( x \in [0, \pi] \), \( f(x) = x \).
- For \( x \in [\pi, 2\pi] \), \( f(x) = 2\pi - x \).

### Step 3: Analyze the right-hand side of the equation

The right-hand side of the equation is:

\[ y = \frac{10 - x}{10} \]

This is a linear equation with a slope of \(-\frac{1}{10}\) and a y-intercept of \(1\).

### Step 4: Find intersections in the interval \([0, 2\pi]\)

1. **For \( x \in [0, \pi] \):**
- Set \( f(x) = x \):
\[ x = \frac{10 - x}{10} \]
- Rearranging gives:
\[ 10x = 10 - x \implies 11x = 10 \implies x = \frac{10}{11} \]
- Since \( \frac{10}{11} \) is in the interval \([0, \pi]\), this is one solution.

2. **For \( x \in [\pi, 2\pi] \):**
- Set \( f(x) = 2\pi - x \):
\[ 2\pi - x = \frac{10 - x}{10} \]
- Rearranging gives:
\[ 10(2\pi - x) = 10 - x \implies 20\pi - 10x = 10 - x \implies 20\pi - 10 = 9x \implies x = \frac{20\pi - 10}{9} \]
- Check if \( \frac{20\pi - 10}{9} \) is in the interval \([\pi, 2\pi]\):
- \( \pi \leq \frac{20\pi - 10}{9} \leq 2\pi \) simplifies to \( 9\pi \leq 20\pi - 10 \) and \( 20\pi - 10 \leq 18\pi \), both of which hold true. Thus, this is another solution.

### Step 5: Extend to the interval \([2\pi, 4\pi]\)

The function \( f(x) \) is periodic, so the same analysis applies:

1. **For \( x \in [2\pi, 3\pi] \):**
- The same equation \( f(x) = x \) will yield the same solution \( x = \frac{10}{11} + 2\pi \).

2. **For \( x \in [3\pi, 4\pi] \):**
- The same equation \( f(x) = 2\pi - x \) will yield the same solution \( x = \frac{20\pi - 10}{9} + 2\pi \).

### Conclusion

Thus, we have found a total of 4 solutions in the interval \([0, 4\pi]\):

1. \( x = \frac{10}{11} \)
2. \( x = \frac{20\pi - 10}{9} \)
3. \( x = \frac{10}{11} + 2\pi \)
4. \( x = \frac{20\pi - 10}{9} + 2\pi \)

Therefore, the number of points \( x \) in \([0, 4\pi]\) satisfying the equation is **4**.