Question

If $f\left(xy\right)=f\left(x\right)\cdot f\left(y\right)$ and $f\left(3\right)=1$, then $f^{\prime }\left(10\right)$ is equal to ________.

• A. $10$
• B. $1$
• C. $0$
• D. none

Answer 1

Answer: (C)

$0$

Given $f\left(xy\right)=f\left(x\right)\cdot f\left(y\right)$

$f\left(3\right)=1$

let $x=3$, and $y=1$,we get

$f\left(3\right)=f\left(3\right)\cdot f\left(1\right)$

$f\left(1\right)=1$

and if $x=3,andy=\frac{1}{3}$

then we also get $f\left(\frac{1}{3}\right)=1$

so this is a constant function

so $f^{\prime }\left(10\right)=0$