Let f be a real valued function defined on the interval \left(\right. 0 , \infty \left.\right) by \textrm{ } f \left(\right. x \left.\right) = lnx + \textrm{ } \int_{0}^{x} \sqrt{1 + sint} \textrm{ } dt . Then which of the following statement(s) is/are true?

Engineering

Mathematics

Maxima and Minima

Question

Let f be a real valued function defined on the interval $(0, \infty) by f (x) = lnx + \int_{0}^{x} \sqrt{1 + sint} dt$ . Then which of the following statement(s) is/are true?

f '' (x) exists for all x $\in$ (0, $\infty$ )

f ' (x) exists for all x $\in$ (0, $\infty$ ) and f ' is continuous on (0, $\infty$ ) but not differentiable on (0, $\infty$ ).

there exists $β$ > 0 such that | f (x) | + | f ' (x) | $\leq β$ b for all x $\in$ (0, $\infty$ )

there exists $α$ > 1 such that | f ' (x) | < | f (x) | for all x $\in$ (a, $\infty$ )

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$f (x) = \ln x + \int_{. 0}^{x} \sqrt{1 + \sin t d t}$

$f' (x) = \frac{1}{x} + \sqrt{1 + \sin x} = \frac{1}{x} + | \cos \frac{x}{2} + \sin \frac{x}{2} |$

$\frac{1}{x} + \sqrt{1 + \sin x} < \ln x + \int_{0}^{x} \sqrt{1 + \sin t} dt$

$∵ lnx > \frac{1}{x} for some x = α \forall α > 1$

$and \sqrt{1 + \sin x} < \int_{0}^{x} \sqrt{1 + \sin t dt}$

For some x = $α \forall α$ > 1