Let `F:RtoR` be a thrice differntiable function. Suppose that `F(1)=0,F(3)=-4` and `F'(x)lt0` for all `x epsilon(1//2,3)`. Let `f(x)=xF(x)` for all `x inR`. Then the correct statement(s) is (are)

Options

A. a. `f'(1)lt0`
B. b. `f(2)lt0`
C. c. `f'(x)!=0` for an `x in (1,3)`
D. d. `f'(x)=0` for some `x in (1,3)`

Explanation

To solve the problem step by step, we will analyze the given function \( F: \mathbb{R} \to \mathbb{R} \) and the derived function \( f(x) = x F(x) \).

### Step 1: Understanding the given information
We are given:
- \( F(1) = 0 \)
- \( F(3) = -4 \)
- \( F'(x) < 0 \) for all \( x \in \left(\frac{1}{2}, 3\right) \)

This implies that \( F(x) \) is a decreasing function in the interval \( \left(\frac{1}{2}, 3\right) \).

### Step 2: Finding \( f'(x) \)
We need to differentiate \( f(x) = x F(x) \) using the product rule:
\[ f'(x) = F(x) + x F'(x) \]

### Step 3: Evaluating \( f'(1) \)
Substituting \( x = 1 \) into the derivative:
\[ f'(1) = F(1) + 1 \cdot F'(1) = 0 + F'(1) = F'(1) \]
Since \( F'(x) < 0 \) for \( x \in \left(\frac{1}{2}, 3\right) \), it follows that:
\[ f'(1) < 0 \]
Thus, option (A) is correct: \( f'(1) < 0 \).

### Step 4: Evaluating \( f(2) \)
Next, we evaluate \( f(2) \):
\[ f(2) = 2 F(2) \]
Since \( F(x) \) is decreasing and \( F(1) = 0 \) and \( F(3) = -4 \), it follows that:
\[ F(2) < F(1) = 0 \quad \text{and} \quad F(2) > F(3) = -4 \]
Thus, \( F(2) < 0 \), which implies:
\[ f(2) = 2 F(2) < 0 \]
So, option (B) is correct: \( f(2) < 0 \).

### Step 5: Analyzing \( f'(x) \) for \( x \in (1, 3) \)
We know:
\[ f'(x) = F(x) + x F'(x) \]
For \( x \in (1, 3) \):
- Since \( F(x) < 0 \) (as \( F(1) = 0 \) and \( F(3) = -4 \) and \( F(x) \) is decreasing),
- And \( F'(x) < 0 \), thus \( x F'(x) < 0 \) (since \( x > 0 \)).

Therefore, both terms in \( f'(x) \) are negative:
\[ f'(x) < 0 \quad \text{for } x \in (1, 3) \]
This means \( f'(x) \neq 0 \) for \( x \in (1, 3) \), so option (C) is correct.

### Step 6: Conclusion
Since all the options (A), (B), and (C) are correct, we conclude that options (A), (B), and (C) are true.