Question

Let $f(x + y) = f(x) f(y)$ for all $x$ and $y$. If $f(0) = 1, f(3) = 3$ and $f ′(0) =11,$ then $f ′(3)$ is equal to

• A. 11
• B. 22
• C. 33
• D. 44
• E. 55

Answer 1

Answer: (C)

We have
$ f^{'}(x) =\displaystyle\lim_{h \rightarrow 0} \frac{f(x +h)-f(x)}{h} $
$\Rightarrow f^{'}(3) =\displaystyle\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h} $
$=\displaystyle \lim _{h \rightarrow 0} \frac{f(3) f(h)-f(3+0)}{h} $
$=\displaystyle \lim_{h \rightarrow 0} \frac{f(3) f(h)-f(3) f(0)}{h}$
$=f(3)\displaystyle \lim_{h \rightarrow 0} \frac{f(h)-f(0)}{h} $
$=f(3)\displaystyle \lim_{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$
$=f(3) f^{'}(0) $
$=3 \times 11 $
$=33$