Question

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

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

Answer 1

Answer: (D)

We have,
$f(x+y)=f(x)f(y)$
differentiate with respect to $x,y$ as constant, we get
$f^{\prime }(x+y)=f^{\prime }(x)f(y)$
Put $x=0,y=3$, we get
$f^{\prime }(0+3)=f^{\prime }(0)f(3)$
$\Rightarrow f^{\prime }(3)=f^{\prime }(0)\cdot f(3)$
$\Rightarrow f^{\prime }(3)=11\times 3=33$
$[\because f^{\prime }(0)=11,f(3)=3]$