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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2016 · Mathematics · Continuity and Differentiability

Paragraph: Let f(x)=\ array ll -2, & -3 ≤ x ≤ 0 x-2, & 0 Question:...

Paragraph: Let \mathrm{f}(\mathrm{x})=\left\{−2,−3≤x≤0x−2,0<x≤3\begin{array}{ll}-2, & -3 \leq \mathrm{x} \leq 0 \\ \mathrm{x}-2, & 0 < \mathrm{x} \leq 3\end{array}\right. and g(x)=f(∣x∣)+∣f(x)∣\mathrm{g}(\mathrm{x})=\mathrm{f}(|\mathrm{x}|)+|\mathrm{f}(\mathrm{x})|
Question: Which of the following statements are correct? 1. g(x)\mathrm{g}(\mathrm{x}) is continuous at x=0\mathrm{x}=0. 2. g(x)g(x) is continuous at x=2x=2. 3. g(x)\mathrm{g}(\mathrm{x}) is continuous at x=−1\mathrm{x}=-1. Select the correct answer using the code given below:

Options

  1. A1 and 2 only
  2. B2 and 3 only
  3. C1 and 3 only
  4. D1,2 and 3

Correct answer

D. 1,2 and 3

Step-by-step solution

At x=0x=0 For LHL: g(x)=−2+∣−2∣=0g(x)=-2+|-2|=0 For RHL: g(x)=∣x∣−2+∣x−2∣g(x)=|x|-2+|x-2|g(x)=x−2−(x−2)=0g(x)=0\begin{aligned} g(x) & =x-2-(x-2)=0 \\ g(x) & =0 \end{aligned}For $(x=0): g(x)=-2+|-2|=0\begin{aligned} & \mathrm{LHL}=\lim _{x \rightarrow 0^{-}} g(x)=0 \\ & \mathrm{RHL}=\lim _{x \rightarrow 0^{+}} g(x)=0 \\ & g(0)=0 \end{aligned}\Rightarrow g(x)$ is continuous at $x=0$ At $x=2$ For LHL: $g(x)=|x|-2+|x-2|\begin{aligned} & g(x)=x-2-(x-2) \\ & g(x)=0 \end{aligned}$ For RHL: $g(x)=|x|-2+|x-2|\begin{aligned} & g(x)=x-2+x-2 \\ & 2(x)=2 x-4 \end{aligned}$ For $(x=2): g(x)=|x|-2+|x-2|\mathrm{LHL}=\lim _{x \rightarrow 2^{-}} g(x)=0\begin{aligned} & \mathrm{RHL}=\lim _{x \rightarrow 2^{+}} g(x)=\lim _{x \rightarrow 2} 2 x-4=2(2)-4=0 \\ & g(2)=|2|-2+|2-2| \\ & g(2)=2-2+2-2=0 \end{aligned}\Rightarrow g(x)$ is continuous at $x=2$. At $x=-1$ For LHL: $g(x)=-2+|-2|=0$ For RHL: $g(x)=-2+|-2|=0$ For $(x=-1): g(x)=-2+|-2|=0\begin{aligned} & \therefore \mathrm{LHL}=\lim _{x \rightarrow-1^{-}} g(x)=0 \\ & \mathrm{RHL}=\lim _{x \rightarrow-1^{+}} g(x)=0 \\ & g(-1)=0 \end{aligned}$⇒g(x)\Rightarrow g(\mathrm{x}) is differentiable at x=−1x=-1.
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