Prove that function `f(x)=cos sqrt(x)` is non-periodic.
Lost your password? Please enter your email address. You will receive a link and will create a new password via email.
Please briefly explain why you feel this question should be reported.
Please briefly explain why you feel this answer should be reported.
Please briefly explain why you feel this user should be reported.
We have `f(x)=cos sqrt(x)`
Let f(x) be periodic with period T, where `T gt 0.`
` :. f(x+T)=f(x)`
`implies cos sqrt(x+T)=cos sqrt(x) ” for ” x ge 0.`
In particular choosing ` x=0`, we have
`cos sqrt(T)=cos sqrt(0)=1 ” …(1)” `
For `x=T`, we have
`cos sqrt(T+T)=cos sqrt(T)=1`
or `cos sqrt(2T)=1 ” …(2)” `
From (1) , `sqrt(T) =2m pi, m in Z`
From (2), `sqrt(2T)=2n pi, n in Z`
` :. (sqrt(2T))/(sqrt(T))=(2n pi)/(2m pi)`
or ` sqrt(2)=(n)/(m),` which is not true.
So, `cos sqrt(x)` is not periodic.