Question

Let $f(x+y)=f\left(x\right)f\left(y\right)\forall x,y\in R,f\left(5\right)=2,f^{\prime }\left(0\right)=3$, then $f^{\prime }\left(5\right)$ equals:

• A. $4$
• B. $1$
• C. $\frac{1}{2}$
• D. $6$

Answer 1

The correct option is D $6$

When $f(x+y)=f\left(x\right)f\left(y\right)$, then $f\left(x\right)=a^{cx}$

Now, $f\left(5\right)=a^{5c}$

$\Rightarrow c=\frac{{\log }_{a}2}{5}$

Now,$f^{\prime }\left(x\right)=c{a}^{cx}\ln a$

$\Rightarrow f^{\prime }\left(0\right)=c\ln a$

$\Rightarrow 3=c\ln a$

and $f^{\prime }\left(5\right)=\frac{3}{\ln a}.a^{{\log }_{a}2}.\log a$

$f^{\prime }\left(5\right)=3\times 2$

$f^{\prime }\left(5\right)=6$