$\text{ If }f\left(x\right)f\left(y\right)=f(x+y)\text{ for all }x,y,\text{ suppose }f\left(5\right)=2\text{ and }{f}^1\left(0\right)=3\text{, }$$thenf\left(5\right)isequalto$

$\begin{array}{l}f(x+y)=f\left(x\right)f\left(y\right)\\ \Rightarrow f(0+5)=f\left(0\right)f\left(5\right)\Rightarrow f\left(0\right)=1\left[\therefore f\left(5\right)=2\ne 0\right]\\ Also,f\left(5\right)=\underset{h\to 0}{\lim }\frac{f(5+h)-f(5+0)}{h}\\ =\underset{h\to 0}{\lim }\frac{f\left(5\right)f\left(h\right)-f\left(5\right)f\left(0\right)}{h}\\ =f\left(5\right)\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h}\\ =f\left(5\right)f^1\left(0\right)=2\times 3=6\\ Hence\left(1\right)isthecorrectanswer.\end{array}$