Question

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

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

Answer 1

Answer: (C)

We have
$f^{{}^{\prime }}(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$
$\Rightarrow f^{{}^{\prime }}(3)=\underset{h\to 0}{\lim }\frac{f(3+h)-f(3)}{h}$
$=\underset{h\to 0}{\lim }\frac{f(3)f(h)-f(3+0)}{h}$
$=\underset{h\to 0}{\lim }\frac{f(3)f(h)-f(3)f(0)}{h}$
$=f(3)\underset{h\to 0}{\lim }\frac{f(h)-f(0)}{h}$
$=f(3)\underset{h\to 0}{\lim }\frac{f(0+h)-f(0)}{h}$
$=f(3)f^{{}^{\prime }}(0)$
$=3\times 11$
$=33$