#### Passage VI (Q

MATHEMATICS · Single correct · AoPS · Mathematics · Functions · Functions

Problem

#### Passage VI (Q. Nos. 105 to 106) If

f: \mathbb{R} \to (0, \infty)

is a differentiable function

f(x)

satisfying

f(x+y) - f(x-y) = f(x) \cdot \{f(y) - f(-y)\}, \forall x, y \in \mathbb{R}

(f(y) \neq f(-y) \text{ for all } y \in \mathbb{R})

and

f'(0) = 2010

. Now, answer the following questions: Which of the following is true for

f(x)

? (1)

f(x)

is one-one and into (2)

\{f(x)\}

is non-periodic, where

\{\cdot\}

denotes fractional part of

x

(3)

f(x) = 4

has only two solutions (4)

f(x) = f'(x)

has only one solution