1thank u sir...
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...
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.
62@arshad.. for polynomials these wil always hold.. (sorry i made that assumption :P)
but even otherwise try to prove these ...