To solve the problem, we need to analyze the function \( f(x) \) defined as:
\[
f(x) = \int_a^x g(t) \, dt
\]
Given that \( f(x) \) has 5 roots between \( (a, b) \), we want to find the number of roots of the product \( g(x)g'(x) \) in the same interval.
### Step-by-Step Solution:
1. **Differentiate \( f(x) \)**:
By applying the Fundamental Theorem of Calculus, we differentiate \( f(x) \):
\[
f'(x) = g(x)
\]
2. **Roots of \( f'(x) \)**:
Since \( f(x) \) has 5 roots, it means that \( f(x) = 0 \) at 5 distinct points in the interval \( (a, b) \). By Rolle's Theorem, between any two successive roots of \( f(x) \), there exists at least one root of \( f'(x) \).
- If we denote the roots of \( f(x) \) as \( r_1, r_2, r_3, r_4, r_5 \), then by Rolle's Theorem, there will be 4 roots of \( f'(x) \) (since there are 5 intervals between the 5 roots).
3. **Differentiate \( f'(x) \)**:
Now, we differentiate \( f'(x) \) to find \( f''(x) \):
\[
f''(x) = g'(x)
\]
4. **Roots of \( f''(x) \)**:
Similarly, between each pair of roots of \( f'(x) \), there will be at least one root of \( f''(x) \). Since \( f'(x) \) has 4 roots, there will be 3 roots of \( f''(x) \) (since there are 4 intervals between the 4 roots).
5. **Finding Roots of \( g(x)g'(x) \)**:
The expression \( g(x)g'(x) = 0 \) implies either \( g(x) = 0 \) or \( g'(x) = 0 \).
- From the previous steps, we found that \( g(x) = 0 \) has 4 roots (from \( f'(x) \)).
- \( g'(x) = 0 \) has 3 roots (from \( f''(x) \)).
Since the roots of \( g(x) \) and \( g'(x) \) are distinct (they do not coincide), we can add the number of roots together.
6. **Total Number of Roots**:
Therefore, the total number of roots of \( g(x)g'(x) \) in the interval \( (a, b) \) is:
\[
4 + 3 = 7
\]
### Final Answer:
The number of roots of \( g(x)g'(x) \) between \( (a, b) \) is **7**.
