Let f be a real-valued function defined on the interval (0, ∞) by f (x) = ln x + \int^{x}_{0} \sqrt{1+sint} dt. Then, which of the following statement(s) is(are) true? [JEE Adv 2010 P1]

f ″(x) exists for all x ∈(0,∞).

f ′(x) exists for all x ∈(0,∞) and f′ is continuous on (0, ∞), but not differentiable on (0, ∞).

there exists α > 1 such that |f ′(x)| < |f (x)| for all x ∈(a,∞).

there exists β > 0 such that |f (x)| + |f ′(x)| ≤ β for allx ∈(0,∞).

Application of Integration

Section: Mathematics

Question : 24 of 28

Marks: +1, -0

Let f be a real-valued function defined on the interval (0, ∞) by f (x) = ln x +

∫

√1+sintdt. Then, which of the following statement(s) is(are) true?

[JEE Adv 2010 P1]

f ″(x) exists for all x ∈(0,∞).
f ′(x) exists for all x ∈(0,∞) and f′ is continuous on (0, ∞), but not differentiable on (0, ∞).
there exists α > 1 such that |f ′(x)| < |f (x)| for all x ∈(a,∞).
there exists β > 0 such that |f (x)| + |f ′(x)| ≤ β for allx ∈(0,∞).

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