Let `f (x)=(x+1) (x+2) (x+3)…..(x+100) and g (x) =f (x) f''(x) -f ('(x)) ^(2).` Let n be the numbers of rreal roots of `g(x) =0,` then:

Options

A. `n lt 2`
B. `n gt 2`
C. `n lt 100`
D. `n gt 100`

Answer

To solve the problem, we need to analyze the function \( g(x) = f(x) f''(x) - (f'(x))^2 \) and determine the number of real roots of \( g(x) = 0 \).

### Step 1: Define the function \( f(x) \)
The function \( f(x) \) is given by:
\[ f(x) = (x + 1)(x + 2)(x + 3) \cdots (x + 100) \]
This is a polynomial of degree 100, and it has roots at \( x = -1, -2, -3, \ldots, -100 \).

### Step 2: Differentiate \( f(x) \)
To find \( f'(x) \) and \( f''(x) \), we will use the product rule. However, since \( f(x) \) is a product of linear factors, we can express the derivatives in terms of the roots.

- The first derivative \( f'(x) \) will be a polynomial of degree 99.
- The second derivative \( f''(x) \) will be a polynomial of degree 98.

### Step 3: Analyze \( g(x) \)
The expression for \( g(x) \) is:
\[ g(x) = f(x) f''(x) - (f'(x))^2 \]
This is a polynomial expression. To find the roots of \( g(x) = 0 \), we need to analyze the behavior of \( g(x) \).

### Step 4: Consider the roots of \( f(x) \)
The roots of \( f(x) \) are \( x = -1, -2, \ldots, -100 \). At these points:
- \( f(x) = 0 \)
- Therefore, \( g(x) \) becomes \( 0 \cdot f''(x) - (f'(-k))^2 = - (f'(-k))^2 \) for \( k = 1, 2, \ldots, 100 \).

Since \( f'(-k) \) is not zero (as \( f(x) \) has distinct roots), we conclude that \( g(-k) < 0 \) for each root \( -k \).

### Step 5: Analyze the behavior of \( g(x) \)
Since \( g(x) \) is a polynomial, we can analyze its behavior at the roots of \( f(x) \):
- Between each pair of roots \( -k \) and \( -(k+1) \), \( g(x) \) must cross the x-axis at least once due to the Intermediate Value Theorem.

### Step 6: Count the roots
Given that \( f(x) \) has 100 roots, we can conclude that:
- \( g(x) \) can potentially have roots in the intervals between these roots of \( f(x) \).
- However, since \( g(x) \) is constructed from \( f(x) \) and its derivatives, we need to check if \( g(x) \) can actually equal zero.

### Conclusion
After analyzing \( g(x) \):
- \( g(x) \) does not have any real roots since it is always positive or negative based on the roots of \( f(x) \).
- Therefore, the number of real roots \( n \) of \( g(x) = 0 \) is \( 0 \).

Thus, the final answer is:
\[ n = 0 \]