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Let f(x) be periodic and k be a positive...

Let `f(x)` be periodic and `k` be a positive real number such that `f(x+k)+f(x)=0fora l lx in Rdot` Prove that `f(x)` is periodic with period `2kdot`

A

k

B

2k

C

`(k)/(2)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
We have,
f(X+k)+f(x)=0 for all ` x in R `
`implies f(x+k)=-f(x) ` for all ` x in R`
`f(x+2k)=-f(x+k)` for al ` x in R `
`implies f(x+2k)=f(x) ` for all ` x in R " " [ :' f(x+k)=-f(x)]`
Hence, f(x) is a periodic function with period 2k.
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OBJECTIVE RD SHARMA ENGLISH-REAL FUNCTIONS -Chapter Test
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