Let f(x) = cos px + sin x be a periodic funct | vedclass

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(A) function $f(x) = \cos px + \sin x$ is periodic if there exists a period $\lambda > 0$ such that $f(x + \lambda) = f(x)$ for all $x \in R$.This imp

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Rational

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(A) function $f(x) = \cos px + \sin x$ is periodic if there exists a period $\lambda > 0$ such that $f(x + \lambda) = f(x)$ for all $x \in R$.This implies $\sin(x + \lambda) + \cos p(x + \lambda) = \sin x + \cos px$.For this to hold,both $\sin x$ and $\cos px$ must be periodic with a common period $\lambda$.The period of $\sin x$ is $2\pi$,and the period of $\cos px$ is $\frac{2\pi}{|p|}$.For the sum to be periodic,the ratio of the periods must be a rational number:$\frac{2\pi}{2\pi/|p|} = |p| \in Q$.Thus,$p$ must be a rational number.

Let $f(x) = \cos px + \sin x$ be a periodic function,then $p$ must be

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