Let `f(x)= (x-2) (x -3) (x-4) (x-5) (x-6)` then

Options

A. f'(x) = 0 has five real roots
B. four roots of f'(x) = 0 lie in `(2, 3) uu (3,4) uu (4,5) uu (5,6)`
C. the equation f'(x) has only three roots
D. four roots of f'(x) = 0 lie in `(1,2) uu (2,3) uu(3,4) uu (4,5)`

Answer

To solve the problem, we need to analyze the function \( f(x) = (x-2)(x-3)(x-4)(x-5)(x-6) \) and find the number of roots of its derivative \( f'(x) \).

### Step 1: Identify the roots of \( f(x) \)
The function \( f(x) \) is a polynomial of degree 5, and its roots are clearly visible from the factors:
- The roots are \( x = 2, 3, 4, 5, 6 \).

### Step 2: Determine the behavior of \( f(x) \)
Since \( f(x) \) is a polynomial, it is continuous and differentiable everywhere. The polynomial will change its sign at each root:
- For \( x < 2 \), all factors \( (x-2), (x-3), (x-4), (x-5), (x-6) \) are negative, so \( f(x) < 0 \).
- For \( 2 < x < 3 \), \( (x-2) \) is positive and the rest are negative, so \( f(x) > 0 \).
- For \( 3 < x < 4 \), \( (x-2) \) and \( (x-3) \) are positive, and the rest are negative, so \( f(x) < 0 \).
- For \( 4 < x < 5 \), \( (x-2), (x-3), (x-4) \) are positive, and the rest are negative, so \( f(x) > 0 \).
- For \( 5 < x < 6 \), \( (x-2), (x-3), (x-4), (x-5) \) are positive, and \( (x-6) \) is negative, so \( f(x) < 0 \).
- For \( x > 6 \), all factors are positive, so \( f(x) > 0 \).

### Step 3: Find the derivative \( f'(x) \)
To find the critical points where \( f'(x) = 0 \), we note that the derivative will be zero at the local maxima and minima of \( f(x) \). Since \( f(x) \) is a 5th degree polynomial, it can have up to 4 critical points (local maxima and minima).

### Step 4: Analyze the critical points
The critical points will occur between the roots of \( f(x) \):
- The local maxima and minima will occur between the intervals:
- Between \( 2 \) and \( 3 \)
- Between \( 3 \) and \( 4 \)
- Between \( 4 \) and \( 5 \)
- Between \( 5 \) and \( 6 \)

Thus, there are 4 intervals where the slope of the tangent (i.e., \( f'(x) \)) can be zero.

### Conclusion
The number of roots of \( f'(x) \) is 4.