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Prove that function `f(x)=cos sqrt(x)` is non-periodic. |
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Answer» 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. |
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