Calc 4 question, please help

Question

Suppose f : R → (0,∞) is a differentiable function such that

 

f(x + y) = f(x)f(y)     (x, y ∈ R)

 

Show that there is a constant a ∈ R such that

 

f(x) = e^ax

Answer 1

If f(x) = e^ax, then f(x+y) = e^a(x+y)

 

                                     = e^ax+ay = e^axe^ay

 

                                                   = f(x)f(y)

 

f(x+y) = f(x)f(y) for any real number, a.