Q 1629580411.     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?

JEE 2010 ADVANCED Paper 1

(This question may have multiple correct answers)

A `f''(x) ` exists for all `x\in(0,\ \infty)`
B `f(x)` exists for all `x\in(0,\ \infty) ` and `f'` is continuous on `(0,\ \infty)`, but not differentiable on `(0,\ \infty)`
C There exists `\alpha > 1` such that `|f'(x)| < |f(x)|` for all `x\in(\alpha,\ \infty)`
D There exists `\beta > 0` such that `|f(x)|+|f'(x)|\le\beta` for all `x\in(0,\ \infty)`

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