66These statements are not true always. They require some additional constraints. The first four require the continuity of f at atleast one point or monotonocity. The last one require that f(x) be a polynomial.
1)f(xy)=f(x)+f(y) implies f(x)=klnx or f(x)=0
2)f(xy)=f(x).f(y) implies f(x)=x^n
3)f(x+y)=f(x).f(y) implies f(x)=a^kx
4)f(x+y)=f(x)=(fy) implies f(x)=k where k is a constant
and
5)f(x).f(1/x)=f(x)+f(1/x) implies f(x)=±x^n +1
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6 Answers
1the last one is well proved !
i know you have taken this extract from "differential calculus "---arihant !
62Arshad this is a good set of quesitons to prove...
THe last one was proved by Prophet sir here if i remember correctly...
The others are a good exercise.. (Dont learn these results directly!)
1The first three can also be proven. If you want, i'll let you know the proof...
66
62@arshad.. for polynomials these wil always hold.. (sorry i made that assumption :P)
but even otherwise try to prove these ...