Let f be a real-valued function defined on interval $\left(0,\infty \right)$,by $f\left(x\right)=\ln x+{\int }_0^x\sqrt{1+\sin t}.dt$. Then which of the following statement(s) is (are) true? (A). f"(x) exists for all $\in$ $\left(0,\infty \right)$.    (B). f'(x) exists for all x $\in$ $\left(0,\infty \right)$ and f' is continuous on $\left(0,\infty \right)$, but not differentiable on $\left(0,\infty \right)$.    (C). there exists $\alpha >1$ such that $\left|f^{\prime }\left(x\right)\right|<\left|f\left(x\right)\right|$ for all x $\in$ $\left(\alpha ,\infty \right)$.    (D). there exists $\beta >1$ such that $\left|f\left(x\right)\right|+\left|f^{\prime }\left(x\right)\right|\le \beta$ for all x $\in$ $\left(0,\infty \right)$.

Views: 792 students