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Let f(x) be a deriviable function at x=0 and f {(x+y) / k } = f { (x) + f(y) } / k  for ( k belonging to R and k ≠ 0,2) Then show that f(x) is either zero or odd function.

8 years ago

147 Points

Dear Sanchit Gupta

Given

f {(x+y) / k } = {f  (x) + f(y) } / k

Put x=y=0

f(0)=(2/k)f(0)

f(0) [k-2]/k =0

but can not be 2

so f(0) =0

now put y=-x

f(0) =1/k{f(x)+f(-x)}

or   f(-x)=-f(x)

f(x) is a odd function

Please feel free to post as many doubts on our discussion forum as you can. If you find any question Difficult to understand - post it here and we will get you the answer and detailed solution very quickly.

All the best Sanchit.

Regards,

8 years ago
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