For the function \( f(x)=x \cos \frac{1}{x}, x \geq 1 \), [More than One Correct Option, (a) for...

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For the function \( f(x)=x \cos \frac{1}{x}, x \geq 1 \),
[More than One Correct Option,
(a) for at least one \( x \) in the interval \( [1, \infty), f(x+2)-f(x)<2 \)
(b) \( \lim _{x \rightarrow \infty} f^{\prime}(x)=1 \)
(c) for all \( x \) in the interval \( [1, \infty), f(x+2)-f(x)>2 \)
(d) \( f^{\prime}(x) \) is strictly decreasing in the interval \( [1, \infty) \)
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