Question

If $f$ is differentiable, $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x,y\in R,f\left(3\right)=3,f^{\prime }\left(0\right)=11$, then $f^{\prime }\left(3\right)=$

• A. $\frac{3}{11}$
• B. $\frac{11}{3}$
• C. $8$
• D. $33$

Answer 1

The correct option is D $33$

$f(x+y)=f\left(x\right).f\left(y\right)$

Put $y=3$ in the above equation, we get

$f(x+3)=f\left(x\right).f\left(3\right)$

Differentiating it with respect to x,

$f^{\prime }(x+3)=f^{\prime }\left(x\right).f\left(3\right)$

Now putting $x=0$,

=> $f^{\prime }\left(3\right)=f^{\prime }\left(0\right).f\left(3\right)=11\times 3=33$