Question

Let $f(x+y)=f\left(x\right)\cdot f\left(y\right)$ for all $x$ and $y$. If $f\left(3\right)=2$ and $f^{\prime }\left(0\right)=4$, then $f^{\prime }\left(3\right)$ is

• A. $2$
• B. $3$
• C. $8$
• D. $18$

Answer 1

The correct option is C $8$

$f(x+y)=f\left(x\right)\cdot f\left(y\right)....\left(1\right)$
Put $y=3,$ in equation $\left(1\right)$
$f(x+3)=f\left(x\right)\cdot f\left(3\right)f(x+3)=2.f\left(x\right)$
Differentiating both side
$f^{\prime }(x+3)=2f^{\prime }\left(x\right)....\left(2\right)$
Put $x=0,$ in equation $\left(2\right)$
$f^{\prime }(0+3)=2\cdot f^{\prime }\left(0\right)f^{\prime }\left(3\right)=2\cdot 4=8$