Let ` f : (a, b) to R ` be twice differentiable function such that `f(x) = int _(a) ^(x) g (t) dt `
for a differentiable function g(x) . If `f(x) = 0` has exactly five distinct roots in `(a,b)`, then `g(x) g'(x) = 0` has at least

Options

A. twelve roots in (a,b)
B. five roots in (a,b)
C. seven roots in (a,b)
D. three roots in (a,b)

Answer

To solve the problem, we start by analyzing the given function \( f(x) \) and its properties.

### Step 1: Understand the function \( f(x) \)
We are given that:
\[ f(x) = \int_a^x g(t) \, dt \]
where \( g(t) \) is a differentiable function.

### Step 2: Find the first derivative \( f'(x) \)
Using the Fundamental Theorem of Calculus, we differentiate \( f(x) \):
\[ f'(x) = g(x) \]

### Step 3: Find the second derivative \( f''(x) \)
Differentiating \( f'(x) \) gives us:
\[ f''(x) = g'(x) \]

### Step 4: Analyze the roots of \( f(x) = 0 \)
We know that \( f(x) = 0 \) has exactly five distinct roots in the interval \( (a, b) \). Let's denote these roots as \( r_1, r_2, r_3, r_4, r_5 \).

### Step 5: Apply Rolle's Theorem
According to Rolle's Theorem, between any two distinct roots of \( f(x) \), there must be at least one root of \( f'(x) \). Since there are five roots of \( f(x) \), there will be at least \( 5 - 1 = 4 \) roots of \( f'(x) \) in the interval \( (a, b) \).

### Step 6: Analyze the roots of \( f'(x) = 0 \)
Since \( f'(x) = g(x) \), we have:
\[ g(x) = 0 \]
This means that \( g(x) \) has at least 4 roots in \( (a, b) \).

### Step 7: Analyze the roots of \( g'(x) = 0 \)
Using the same reasoning, between each pair of roots of \( g(x) = 0 \), there must be at least one root of \( g'(x) = 0 \). Therefore, for the 4 roots of \( g(x) \), there will be at least \( 4 - 1 = 3 \) roots of \( g'(x) \) in the interval \( (a, b) \).

### Step 8: Total roots of \( g(x) g'(x) = 0 \)
Now, we combine the roots:
- From \( g(x) = 0 \), we have at least 4 roots.
- From \( g'(x) = 0 \), we have at least 3 roots.

Thus, the total number of roots of the equation \( g(x) g'(x) = 0 \) is:
\[ 4 + 3 = 7 \]

### Conclusion
Therefore, \( g(x) g'(x) = 0 \) has at least 7 roots in the interval \( (a, b) \).

### Final Answer:
The correct option is **7 roots in \( (a, b) \)**.

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