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Question

Let f be a real-valued function defined on the interval (0,) by f(x)=lnx+x01+sint dt. Then which of the following statement(s) is/are true?

A
There exists α>1 such that |f(x)|<|f(x)| for all x(α,)
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B
f(x)exists for all x(0,) and f is continuous on (0,), but not differentiable on (0,)
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C
There exists β>0 such that |f(x)|+|f(x)|β for all x(0,)
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D
f′′(x)exists for all x(0,)
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Solution

The correct option is B f(x)exists for all x(0,) and f is continuous on (0,), but not differentiable on (0,)
f(x)=1x+1+sinx,x>0 but not diffrentiable in (0,) as sinx may be 1 and then f′′(x) can't exist.
Clearly f(x) is continuous on (0,)
(4) Is not true because f(x) tends to as x tends to
(3) Is true because |f(x)3| if x>1
But |f(x)|>3, if x>e3
So we may take α=e3

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