Is a function periodic f(x) = \cos (x) +\cos(x^2) This one

Deven Livingston 2022-04-24 Answered
Is a function periodic f(x)=cos(x)+cos(x2)
This one I gave my students today, nobody solve it.
Is a following function periodic f:RR
f(x)=cos(x)+cos(x2)
If someone is interested I can show a solution later.
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Answers (2)

veceritzpzg
Answered 2022-04-25 Author has 16 answers

Without using the derivative, the equation f(x)=f(0) has only one solution. Indeed, if cos(x)+cos(x2)=2 then cos(x)=1=cos(x2) so there exists p,qZ such that x=2pπ and x2=2qπ, so π=q2p2. But π is not rational ; absqrt.
This may be overkill, but at least the same reasoning can prove the following : given any β-periodic function g with βRQ and such that g(x)g(0) for x]0,β[, the function xg(x)+g(x2) is not periodic. For instance, this is also true if g is the Weierstrass function (if b2, with the definition used by wiki).

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kayuukor9c
Answered 2022-04-26 Author has 13 answers
If f(x)=cosx+cos(x2) is periodic, then so is f(x)=sinx2xsin(x2), which is impossible since f'(x) is not bounded. (For xn=(2n+12)π (n>0), we have f(xn)=sinxn2xn122nπ which tends to as n.)
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