Let be a real valued function defined on the interval by

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let ff be a real valued function defined on the interval (0,∞)(0, \infty) by f(x)=ln⁡x+∫0x1+sin⁡tdtf(x) = \ln x + \int_{0}^{x} \sqrt{1+\sin t} dt . Then, which of the following statement(s) is/are true? (1) f′′(x)f''(x) exists for all x∈(0,∞)x \in (0, \infty) (2) f′(x)f'(x) exists for all x∈(0,∞)x \in (0, \infty) and f′f' is continuous on (0,∞)(0, \infty) but not differentiable on (0,∞)(0, \infty) (3) There exists α>1\alpha > 1 such that ∣f′(x)∣<∣f(x)∣|f'(x)| < |f(x)| for all x∈(0,∞)x \in (0, \infty) (4) There exists β>0\beta > 0 such that ∣f(x)∣+∣f′(x)∣≤β|f(x)| + |f'(x)| \le \beta from all x∈(0,∞)x \in (0, \infty)

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