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JEE Advanced 2010Paper 1
Exam
JEE Advanced
Year
2010
Paper
Paper 1
Subject
Mathematics
Chapter
Continuity and Differentiability

JEE Advanced 2010 · Paper 1 · Mathematics · Continuity and Differentiability

Let f be a real-valued function defined on the interval (0, ∞) , by...

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} d t. Then which of the following statement(s) is (are) true ?

Options

  1. Af′′(x)f^{\prime \prime}(x) exists for all x∈(0,∞)x \in(0, \infty)
  2. Bf′(x)f^{\prime}(x) exists for all x∈(0,∞)x \in(0, \infty) and f′f^{\prime} is continuous on (0,∞)(0, \infty), but not differentiable on (0,∞)(0, \infty)
  3. Cthere exists α>1\alpha>1 such that \left|f^{\prime}(x)\right| < |f(x)|forallfor allx \in(\alpha, \infty)
  4. Dthere exists β>0\beta>0 such that ∣f(x)∣+∣f′(x)∣≤β|f(x)|+\left|f^{\prime}(x)\right| \leq \beta \quad from all x∈(0,∞)x \in(0, \infty)

Correct answer

B. f′(x)f^{\prime}(x) exists for all x∈(0,∞)x \in(0, \infty) and f′f^{\prime} is continuous on (0,∞)(0, \infty), but not differentiable on (0,∞)(0, \infty)

Step-by-step solution

Here, f′(x)=1x+1+sin⁡x,x>0f^{\prime}(x)=\frac{1}{x}+\sqrt{1+\sin x}, x>0 but f(x)f(x) is not differentiable in (0,∞)(0, \infty) as sin⁡x\sin x may be −1-1 and then f′′(x)=−1x2+cos⁡x21+sin⁡xf^{\prime \prime}(x)=-\frac{1}{x^2}+\frac{\cos x}{2 \sqrt{1+\sin x}} will not exists. ⇒f′(x)\Rightarrow f^{\prime}(x) is continuous for all x∈(0,∞)x \in(0, \infty) but f′(x)f^{\prime}(x) is not differentiable on (0,∞)(0, \infty). ∴\therefore Option (b) is true. Also, f′(x)≤3\quad f^{\prime}(x) \leq 3, if x>1x>1 and f(x)>3\quad f(x)>3, if x>e3x>e^3 ∴\therefore Let α=e3\alpha=e^3 ⇒\Rightarrow Option (c) is true. (d) is not possible as f(x)→∞f(x) \rightarrow \infty when x→∞x \rightarrow \infty. Hence, (b, c) is the correct option.
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