If f(x + y) = 2 f(x) f(y) for all x, y where f'(0) = 3 and f(4) = 2, then f'(4) is equal to
Answer & explanation
Correct answer: option 2
We have,
$f'(4) =\lim\limits_{h \rightarrow 0} \frac{f(4+h)-f(4)}{h}$
$\Rightarrow f'(4) =\lim\limits_{h \rightarrow 0} \frac{f(4+h)-f(4+0)}{h}$
$\Rightarrow f'(4) =\lim\limits_{h \rightarrow 0} \frac{2 f(4) f(h)-2 f(4) f(0)}{h}$ [∵ f(x + y) =2 f(x) f(y)]
$\Rightarrow f'(4) =\lim\limits_{h \rightarrow 0} 2 f(4)\left(\frac{f(h)-f(0)}{h-0}\right)$
$\Rightarrow f'(4)=4 \lim\limits_{h \rightarrow 0} \frac{f(h)-f(0)}{h-0}=4 f'(0)=4 \times 3=12$