Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f : R \to R$ be a continuous function such that $f(x+y) = f(x) + f(y) + f(x).f(y), \forall x, y \in R$. Also $f'(0) = 1$. Then $\left[\frac{f(4)}{f(2)}\right]$ equals ([·] represents greatest integer function)
Step-by-Step Solution
Key Concept: Transform the functional equation by substitution to recognize it as Cauchy's exponential equation $g(x+y) = g(x)g(y)$.
Given the functional equation $1 + f(x+y) = f(x) + f(y) + f(x)f(y) + 1$, substitute $1 + f(x) = g(x)$ to transform this into $g(x+y) = g(x)g(y)$, which is Cauchy's exponential functional equation with solution $g(x) = e^{λx}$. Therefore $f(x) = e^{λx} - 1$. From boundary condition $f'(0) = λ - 1$, we get $f(x) = e^x - 1$. Computing $\frac{e^4 - 1}{e^2 - 1} = \frac{e^2 + 1}{1} = e^2 + 1 ≈ 8$.
Correct Answer: [A-q] [B-r] [C-p] [D-t]