Let f be a real-valued function defined on the interval $$(0, \infty)$$ by $$f(x) = \ln x + \int_{0}^{x}\sqrt{1 + \sin t}$$ dt. Then which of the following statement(s) is (are) true ?

A

$$f''(x)$$ exists for all $$x \in (0, \infty)$$

B

$$f'(x)$$ exists for all $$x \in (0, \infty)$$ $$f'$$ is continuous on $$(0, \infty)$$, but not differentiable on $$(0, \infty)$$

C

there exists $$\alpha > 1$$ such that $$\mid f'(x) \mid < \mid f(x) \mid$$ for all $$x \in (\alpha, \infty)$$

D

there exists $$\beta > 0$$ such that $$\mid f(x) \mid + \mid f'(x) \mid \leq \beta$$ for all $$x \in (0, \infty)$$