# A calculus problem by avi solanki

Calculus Level 5

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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