Question

Let $f\left(xy\right)=f\left(x\right)f\left(y\right)$ for all $x,y$ $\in$ $R$. If $f^{\prime }\left(1\right)=2$ and $f\left(2\right)=4$, then $f^{\prime }\left(4\right)$ equal to

• A. $4$
• B. $1$
• C. $\frac{1}{2}$
• D. $8$

Answer 1

Answer: (D)

$8$

$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$

$=\underset{h\to 0}{lim}\frac{f\left(x\right)f(1+\frac{h}{x})-f\left(x\right)}{h}$

$f^{\prime }\left(x\right)=\frac{f\left(x\right)}{x}\left(f^{\prime }\right(1\left)\right)$

$\int \frac{df\left(x\right)}{f\left(x\right)}=\int \frac{2}{x}dx$

$\log f\left(x\right)=2\log x+c$

Given $f\left(2\right)=4$

$\Rightarrow \log 4=2\log 2+c$

$\Rightarrow c=0$

$\Rightarrow \log f\left(x\right)=2\log x$

$\Rightarrow f\left(x\right)=x^2$

$f^{\prime }\left(x\right)=2x$

$f^{\prime }\left(4\right)=8$