Question

Let \( f(x) \) be a real differentiable function such that \( f(0) = 1 \) and \( f(x + y) = f(x)f'(y) + f'(x)f(y) \) for all \( x, y \in \mathbb{R} \). Then \( \sum_{n=1}^{100} \log_e f(n) \) is equal to :

• A. 2384
• B. 2525
• C. 5220
• D. 2406

Hint:

When solving functional equations, try substituting specific values for \( x \) and \( y \) to simplify the problem.

Answer 1

The Correct Option is B

Given the functional equation \(f(x + y) = f(x)f'(y) + f'(x)f(y)\). Substituting \( x = y = 0 \) yields \(f(0) = f(0)f'(0) + f'(0)f(0)\). With \( f(0) = 1 \), we have \(1 = 2f'(0)\), which implies \(f'(0) = \frac{1}{2}\). Substituting \( y = 0 \) into the original equation gives \(f(x) = f(x)f'(0) + f'(x)f(0)\). This simplifies to \(f(x) = \frac{1}{2}f(x) + f'(x)\), leading to \(f'(x) = \frac{1}{2}f(x)\). Solving the differential equation \(f'(x) = \frac{1}{2}f(x)\) results in \(f(x) = e^{x/2}\). Now, we compute the sum: \[ \sum_{n=1}^{100} \log f(n) = \sum_{n=1}^{100} \log e^{n/2} = \sum_{n=1}^{100} \frac{n}{2} \] The sum of the first 100 integers is \( \sum_{n=1}^{100} n = 5050 \). Therefore, the required sum is: \[ \frac{1}{2} \times 5050 = 2525 \] The answer is \( \boxed{2525} \).