Paragraph: Let f(x)=[|x|-|x-1|]^2 Question: Which of the following equations is/are correct? 1. f(-2)=f(5) 2. f^ (-2)+f^ (0.5)+f^ (3)=4 Select the correct answer using the code given below:

Correct answer: A. 1 only Given f(x)=(|x|-|x+1|)^2f(x)=\ array cc 1 & x ≤ 0 (2 x-1)^2 & 0 < x < 1 1 & x ≥ 1 array . For x=-2 aligned & f(x)=1 so f(-2)=1 & For x=5 & f(x)=1 f(5)=1 aligned Hence f(-2)=f(5) Now, for x=-1 aligned & f^ (x)=0 & f^ (-2)=0 aligned For x=0.5f^ (x)=8 f^ (0.5)=8 For x=3 aligned & f^ (x)=0 f^ (3)=0 & f^ (-2)+f^ (0.5)+f^ (3)=8 ≠ 4 aligned Only statement 1 is correct.

NDA 2016

Exam: NDA
Year: 2016
Subject: Mathematics
Chapter: Functions

NDA 2016 · Mathematics · Functions

Paragraph: Let f(x)=[|x|-|x-1|]^2 Question: Which of the following...

Paragraph:

\text{ Let } f(x)=[|x|-|x-1|]^2

Question: Which of the following equations is/are correct? 1.

\mathrm{f}(-2)=\mathrm{f}(5) 2. \mathrm{f}^{\prime \prime}(-2)+\mathrm{f}^{\prime \prime}(0.5)+\mathrm{f}^{\prime \prime}(3)=4

Select the correct answer using the code given below:

Options

A1 only
B2 only
CBoth 1 and 2
DNeither 1 nor 2

Correct answer

A. 1 only

Step-by-step solution

Given

f(x)=(|x|-|x+1|)^2f(x)=\left\{

\begin{array}{cc} 1 & x \leq 0 \\ (2 x-1)^2 & 0 < x < 1 \\ 1 & x \geq 1 \end{array}

\right.

For

x=-2

\begin{aligned} & f(x)=1 \text{ so } f(-2)=1 \\ & \text{ For } x=5 \\ & f(x)=1 \Rightarrow f(5)=1 \end{aligned}

Hence $f(-2)=f(5)$ Now, for $x=-1

\begin{aligned} & f^{\prime \prime}(x)=0 \\ & f^{\prime \prime}(-2)=0 \end{aligned}

$ For $x=0.5f^{\prime \prime}(x)=8 \Rightarrow f^{\prime \prime}(0.5)=8$ For $x=3

\begin{aligned} & f^{\prime \prime}(x)=0 \Rightarrow f^{\prime \prime}(3)=0 \\ & \Rightarrow f^{\prime \prime}(-2)+f^{\prime \prime}(0.5)+f^{\prime \prime}(3)=8 \neq 4 \end{aligned}$$$ Only statement 1 is correct.

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