A function satisfies the equation for all in and for any in

MATHEMATICS · Multi-part · CBSE · Mathematics · Differential Equations · Differential Equations

Problem

A function

f: \mathbb{R} \to \mathbb{R}

satisfies the equation

f(x+y) = f(x) \cdot f(y)

for all

x, y

\mathbb{R}

and

f(x) \neq 0

for any

x

\mathbb{R}

. Let the function be differentiable at

x=0

and

f'(0) = 2

. (1) Show that

f'(x) = 2f(x)

for all

x

R

. (2) Hence, determine

f(x)