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Options

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

Answer

C

Explanation

We have,
`f(x)="In"x+overset(x)underset(0)int sqrt(1+sint)dt`
`rArr f'(x)=(1)/(x)+sqrt(1+sin x)`
`rArr f'(x)` exits for all `x in (0,oo))` and is continuous on `(0,oo)` Clearly, f'(x) is not differentiable at `x=2npi-(pi)/(2),n in N`. So, f'(x) is not differentiable on `(0,oo)`.