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Application of Integration question

2010 · Shift 1 · Q47

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PYQ Library
JEE Advanced
Mathematics
Application of Integration
2010 · Shift 1 · Q47

Application of Integration question

2010 · Shift 1 · Q47

JEE AdvancedMathematicsApplication of IntegrationMultiple correct+3 / −1

Let

f

be a real-valued function defined on the interval

\left( {0,\infty } \right)

by

\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt {1 + \sin t\,} dt.}

then which of the following statement(s) is (are) true?

A
$f''(x)$ exists for all $x \in \left( {0,\infty } \right)$
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 $\,\,\alpha \gt 1$ such that $\left| {f'\left( x \right)} \right| \lt \left| {f\left( x \right)} \right|$ for all $x \in \left( {\alpha ,\infty } \right)\,$
D
there exists $\beta \gt 0$ such that $\left| {f\left( x \right)} \right| + \left| {f'\left( x \right)} \right| \le \beta$ for all $x \in \left( {0,\infty } \right)$

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