Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
If $f\left(\frac{xy}{2}\right) = \frac{f(x).f(y)}{2}, \forall x, y \in R, f(1) = f'(1) = 2$. Then, $\frac{f(3)}{f'(3)}$ is_____.
Step-by-Step Solution
Key Concept: Use the functional property and derivative definition to establish a relationship between $f(x)$ and $f'(x)$ that determines the function.
From the definition of derivative, $f'(x) = \lim_{h \to 0} \frac{f(2x + h) - f(x)}{h} = \lim_{h \to 0} \frac{f(2x+h) - f(1) + f(1) - f(x)}{h}$. Using properties of $f$ and the given condition $f(1) = f'(1)$, simplify to get $\frac{f(x)}{f'(x)} = x$, which yields $\frac{f(x)}{f'(x)} = x = \frac{f(3)}{f'(3)} = 3$.
Correct Answer: 1