if f(x) is differentiable function such that f(1) = sin 1, f (2)= sin 4 and f(3) = sin 9, then the minimum number of distinct roots of f'(x) = 2x `cosx^(2)` in (1,3) is `"_______"`

Answer

To solve the problem, we need to determine the minimum number of distinct roots of the equation \( f'(x) = 2x \cos(x^2) \) in the interval \( (1, 3) \), given the values of the function \( f(x) \) at specific points.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that:
- \( f(1) = \sin(1) \)
- \( f(2) = \sin(4) \)
- \( f(3) = \sin(9) \)

Since \( f(x) \) is differentiable, it is also continuous.

2. **Applying the Mean Value Theorem:**
By the Mean Value Theorem, there exists at least one \( c_1 \in (1, 2) \) such that:
\[ f'(c_1) = \frac{f(2) - f(1)}{2 - 1} = f(2) - f(1) = \sin(4) - \sin(1) \]

Similarly, there exists at least one \( c_2 \in (2, 3) \) such that:
\[ f'(c_2) = \frac{f(3) - f(2)}{3 - 2} = f(3) - f(2) = \sin(9) - \sin(4) \]

3. **Analyzing \( f'(x) = 2x \cos(x^2) \):**
We need to find the roots of the equation \( f'(x) = 2x \cos(x^2) \).

4. **Finding the Values of \( f'(c_1) \) and \( f'(c_2) \):**
- Let \( A = \sin(4) - \sin(1) \)
- Let \( B = \sin(9) - \sin(4) \)

We need to check if \( A \) and \( B \) have different signs to apply the Intermediate Value Theorem.

5. **Evaluating \( A \) and \( B \):**
- Since \( \sin(x) \) is an increasing function in the interval \( (1, 3) \), we can conclude:
- \( A = \sin(4) - \sin(1) > 0 \)
- \( B = \sin(9) - \sin(4) > 0 \)

Thus, both \( A \) and \( B \) are positive.

6. **Finding Distinct Roots:**
Since \( f'(c_1) \) and \( f'(c_2) \) are both positive, we need to analyze the behavior of \( 2x \cos(x^2) \) in the interval \( (1, 3) \).

- \( 2x \) is positive in \( (1, 3) \).
- \( \cos(x^2) \) oscillates between -1 and 1.

We need to find how many times \( 2x \cos(x^2) \) can equal \( A \) and \( B \).

7. **Conclusion on Roots:**
Since \( \cos(x^2) \) oscillates, it will cross the horizontal line \( y = A \) and \( y = B \) at least once each in the interval \( (1, 3) \). Therefore, we can conclude that there are at least two distinct roots of \( f'(x) = 2x \cos(x^2) \) in the interval \( (1, 3) \).

Thus, the minimum number of distinct roots of \( f'(x) = 2x \cos(x^2) \) in \( (1, 3) \) is **2**.