Let f : R → R be a function given by f(x + y) = f(x) f(y) for all x, y ∈ R. If f(x) ≠ 0 for all x ∈ R and f'(0) exists, then f'(x) equals
Answer & explanation
Correct answer: option 2
We have,
f(x + y) = f(x) f(y) for all x, y ∈ R
⇒ f(0) = f(0) f(0) [Replacing x and y both by 0]
⇒ f(0)(f(0) - 1) = 0
⇒ f(0) - 1 = 0
⇒ f(0) = 1
Again,
f(x + y) = f(x) f(y) for all x, y ∈ R
⇒ f(x + h) = f(x) f(h) for all x, h ∈ R
⇒ f(x + h) - f(x) = f(x)(f(h) - 1)
⇒ $\frac{f(x+h)-f(x)}{h}=f(x)\left(\frac{f(h)-1}{h}\right)$
$\Rightarrow \lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=f(x) \lim\limits_{h \rightarrow 0} \frac{f(h)-f(0)}{h}$ [∵ f(0) = 1]
⇒ f'(x) = f(x) f'(0) for all x ∈ R