TITLE: A problem on functional equation $f(x+y)=f(x)f(y)$
QUESTION [5 upvotes]: A problem from IIT 1981

Let $f(x)$ be a function defined for all x∈R such that
  $f(x+y)=f(x)⋅f(y)$ for all x,y∈R. Let $f′(0)=3$ and $f(5)=2$. Find the value
  of $f′(5)$.

The author of http://paramanands.blogspot.in/2010/08/two-problems-from-iit-jee.html#.ViuP3X4rLIU has pointed out that in no way f(x) can be an exponential function.But my question is any such function possible which satisfies the above conditions?Or is the question truly wrong?

REPLY [5 votes]: you can prove, as done in the paper that if $f(x+y)=f(x)f(y)$ (and is derivable in $0$), then $f'(x)=f'(0)f(x)$, here $f'(x)=3f(x)$ and $f(0)=1$. the only solution of this equation is $f(x) = e^{3x}$, and so we cannot have f(5)=2.