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

12

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)]

$\Rightarrow f'(4) =\lim\limits_{h \rightarrow 0} 2 f(4)\left(\frac{f(h)-f(0)}{h-0}\right)$

$\Rightarrow f'(4)=4 \lim\limits_{h \rightarrow 0} \frac{f(h)-f(0)}{h-0}=4 f'(0)=4 \times 3=12$