Choose the correct statement:

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(B) function $f(x)$ is periodic if there exists a positive constant $T$ such that $f(x+T) = f(x)$ for all $x$ in the domain.For $f(x) = x + \sin 2x$:A

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$x + \sin 2x$ is not a periodic function

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(B) function $f(x)$ is periodic if there exists a positive constant $T$ such that $f(x+T) = f(x)$ for all $x$ in the domain.For $f(x) = x + \sin 2x$:As $x 	o \infty$, $f(x) 	o \infty$, because $x$ is a strictly increasing function and $\sin 2x$ is bounded between $-1$ and $1$. Thus, $f(x+T) = x + T + \sin(2(x+T)) = x + \sin 2x$ cannot hold for any $T > 0$. Therefore, $x + \sin 2x$ is not a periodic function.For $g(x) = \cos (\sqrt{x}+1)$:As $x$ increases, the argument $\sqrt{x}+1$ increases, but the rate of change of the argument decreases. For a function to be periodic, the values must repeat at regular intervals. Since the distance between consecutive zeros of $\cos (\sqrt{x}+1)$ is not constant, it is not a periodic function.Thus, both statements $B$ and $D$ are correct. Given the structure, we identify that $x + \sin 2x$ is not periodic and $\cos (\sqrt{x}+1)$ is not periodic.

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