Let \(\displaystyle f(x) = e^{-1/x^2} + \int_0^{\pi x/2} \sqrt{1 + \sin t} \, dt\quad \forall x \in (0, \infty) \), then which of the following statements is true?

**(A):** \(f'\) exists and is continuous \(\forall x \in (0,\infty) \).

**(B):** \(f'' \) exists \(\forall x\in (0,\infty) \).

**(C):** \(f'\) is bounded.

**(D):** There exists \(\alpha > 0 \) such that \( | f(x) | > | f'(x) | \; \forall x \in (\alpha ,\infty) \).

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