Let f be a real-valued function defined on interval `(0,oo)`,by `f(x)=lnx+int_0^xsqrt(1+sint).dt`. Then which of the following statement(s) is (are) true? (A). f"(x) exists for all `in` `(0,oo)`.`" "` (B). f'(x) exists for all x `in` `(0,oo)` and f' is continuous on `(0,oo)`, but not differentiable on `(0,oo)`.`" "` (C). there exists `alpha>1` such that `|f'(x)|<|f(x)|` for all x `in` `(alpha,oo)`.`" "` (D). there exists `beta>1` such that `|f(x)|+|f'(x)|<=beta` for all x `in` `(0,oo)`.

Options

A. f''(x) exists for all `x in (0, oo)`
B. f'(x) exists for all `x in (0, oo)` and f' is continuous on `(0, oo)` but not differentiable on `(0, oo)`
C. There exists `alpha gt 1` such that `|f'(x)| lt |f (x)|` for all `x in (0, oo)`
D. There exists `beta gt 0` such that `|f(x)| + |f'(x)| le beta` from all `x in (0, oo)`

Answer

B, C