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Let f:R to R be a function satisfying f...

Let `f:R to R ` be a function satisfying `f(2-x)=f(2+x) and f(20-x)=f(x) AA x in R.` For this function `f`, answer the following.
If `f(2) ne f(6),` then the

A

fundamental period of `f(x) " is " 1`

B

fundamental period of `f(x) " may be " 1`

C

period of `f(x)` cannot be 1

D

fundamental period of `f(x) " is " 8`

Text Solution

Verified by Experts

The correct Answer is:
C
`f(2-x) =f(2+x) " (1)" `
Replace x by `2-x. " Then " f(x)=f(4-x) " (2)" `
Also, given `f(20-x)=f(x) " (3)" `
From (1) and (2), `f(4-x)=f(20-x).`
Replace x by `4-x. " Then " f(x)=f(x+16).`
Hence, the period of `f(x)` is 16.
If 1 is a period, then `f(x)=f(x+1) AA x in R`
`or f(2)=f(3)=f(4)=f(5)=f(6)`
which contradicts the given hypotheses that `f(2) ne f(6).`
Therefore, 1 cannot be period of `f(x)`.
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