if f(x)=(x -4) (x-5) (x-6) (x-7) then, (A) f'(x) =0 has four roots (B) three roots f'(x) =0 lie in (4,5) `uu(5,6) uu(6,7)` (C) the equation `f'(x) =0` has only one real root. (D) three roots of f'(x) =0 lie in (3,4) `uu (4,5) uu (5,6)`

Options

A. f(x) =0 has four roots
B. three roots f(x) =0 lie in (4,5) `uu(5,6) uu(6,7)`
C. the equation `f(x) =0` has only one real root.
D. three roots of f(x) =0 lie in (3,4) `uu (4,5) uu (5,6)`

Answer

To solve the problem, we need to analyze the function \( f(x) = (x - 4)(x - 5)(x - 6)(x - 7) \) and its derivative \( f'(x) \).

### Step 1: Identify the function and its roots
The function \( f(x) \) is a polynomial of degree 4, and it has roots at \( x = 4, 5, 6, \) and \( 7 \). This means that \( f(x) = 0 \) at these points.

### Step 2: Find the derivative \( f'(x) \)
To find the critical points where \( f'(x) = 0 \), we first need to differentiate \( f(x) \).

Using the product rule:
\[ f'(x) = (x - 5)(x - 6)(x - 7) + (x - 4)(x - 6)(x - 7) + (x - 4)(x - 5)(x - 7) + (x - 4)(x - 5)(x - 6) \]

### Step 3: Analyze the behavior of \( f'(x) \)
Since \( f(x) \) is a polynomial of degree 4, \( f'(x) \) will be a polynomial of degree 3. A cubic polynomial can have at most 3 real roots.

### Step 4: Determine the intervals where \( f'(x) = 0 \)
To find where \( f'(x) = 0 \), we can analyze the intervals created by the roots of \( f(x) \):
- The roots of \( f(x) \) divide the x-axis into intervals: \( (-\infty, 4) \), \( (4, 5) \), \( (5, 6) \), \( (6, 7) \), and \( (7, \infty) \).

### Step 5: Evaluate the sign of \( f'(x) \) in each interval
1. **Interval \( (4, 5) \)**:
- \( f'(x) \) changes from positive to negative, indicating a local maximum.

2. **Interval \( (5, 6) \)**:
- \( f'(x) \) changes from negative to positive, indicating a local minimum.

3. **Interval \( (6, 7) \)**:
- \( f'(x) \) changes from positive to negative, indicating another local maximum.

### Conclusion:
From the analysis, we find that:
- There are three critical points where \( f'(x) = 0 \), which lie in the intervals \( (4, 5) \), \( (5, 6) \), and \( (6, 7) \).

Thus, the correct option is:
**(B)** Three roots \( f'(x) = 0 \) lie in \( (4, 5) \cup (5, 6) \cup (6, 7) \).