JEE Advanced 2018 · Paper 2 · Mathematics Q11

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MathematicsNumerical+3Question 47 of 54

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with $f(0) = 1$ and satisfying the equation $f(x + y) = f(x)f'(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$ . Then, the value of $\log_e(f(4))$ is _____.

Your answer

Answer and worked solution

Official answer: 2

Put $x=y=0$ . $f(0)=2f(0)f'(0)$ , so $f'(0)=\tfrac12$ .

Put $y=0$ .
$f(x)=f(x)f'(0)+f'(x)f(0)=\tfrac12f(x)+f'(x)\implies f'=\tfrac12f.$

Solve. $f(x)=e^{x/2}$ , which does satisfy the original equation. So $\ln f(4)=2$ .

Answer: 2

Topics:Formation and applications Types of functions

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JEE Advanced 2018 · Paper 2, Mathematics question 11