If `a lt b lt c lt d lt e and f(x) =(x-a)^(2) (x-b) (x -c) (x-d) (x-e )` which of the following is true?

Options

A. `f(x) gt 0`, for `(x lt a)` as well as for `(d lt x lt e)`
B. `f(x) lt 0`, for `(a lt x lt b)` as well as for `(d lt x lt e)`
C. `f(x) lt 0`, for `(b lt x lt c)` as well as for `(d lt x lt e)`
D. None of these

Answer

To solve the problem, we need to analyze the function \( f(x) = (x-a)^2 (x-b)(x-c)(x-d)(x-e) \) given the conditions \( a < b < c < d < e \).

### Step-by-Step Solution:

1. **Understanding the Function**: The function \( f(x) \) consists of a squared term \( (x-a)^2 \) and four linear factors \( (x-b), (x-c), (x-d), (x-e) \). The squared term will always be non-negative (i.e., \( \geq 0 \)) for any real value of \( x \).

2. **Identifying Roots**: The roots of the function occur at \( x = a \), \( x = b \), \( x = c \), \( x = d \), and \( x = e \). The squared term at \( x = a \) means that \( f(x) \) will touch the x-axis at \( x = a \) but will not cross it.

3. **Behavior of the Function**:
- For \( x < a \): All terms \( (x-b), (x-c), (x-d), (x-e) \) are negative, thus \( f(x) \) is positive (since \( (x-a)^2 \) is positive and the product of four negative numbers is positive).
- For \( a < x < b \): \( (x-a)^2 \) is positive, \( (x-b) \) is negative, \( (x-c), (x-d), (x-e) \) are also negative. Thus, \( f(x) \) is positive (positive times three negatives).
- For \( b < x < c \): \( (x-a)^2 \) is positive, \( (x-b) \) is positive, \( (x-c) \) is negative, \( (x-d), (x-e) \) are negative. Thus, \( f(x) \) is negative (positive times one positive and three negatives).
- For \( c < x < d \): \( (x-a)^2 \) is positive, \( (x-b) \) is positive, \( (x-c) \) is positive, \( (x-d) \) is negative, \( (x-e) \) is negative. Thus, \( f(x) \) is negative (positive times three positives and one negative).
- For \( d < x < e \): \( (x-a)^2 \) is positive, \( (x-b) \) is positive, \( (x-c) \) is positive, \( (x-d) \) is positive, \( (x-e) \) is negative. Thus, \( f(x) \) is positive (positive times four positives and one negative).
- For \( x > e \): All terms are positive, hence \( f(x) \) is positive.

4. **Conclusion**: From the analysis, we find that:
- \( f(x) \) is negative when \( x \) is between \( b \) and \( c \) and also when \( x \) is between \( c \) and \( d \).
- Therefore, the correct option is that \( f(x) \) is negative when \( x \) is between \( b \) and \( c \) and also between \( d \) and \( e \).

### Final Answer:
The correct statement is: **f(x) is negative when x is between b and c and also when x is between d and e.**