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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Differentiation

NDA 2016 · Mathematics · Differentiation

Let f: R arrow R be a function such that f(x)=x^3+x^2 f^ (1)+x f f^...

Let f:R→R\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} be a function such that f(x)=x3+x2f′(1)+xff′′(2)+f′′′(3)f(x)=x^3+x^2 f^{\prime}(1)+x f f^{\prime \prime}(2)+f^{\prime \prime \prime}(3) for x∈R\mathrm{x} \in \mathrm{R} Consider the following: 1. f(2)=f(1)−f(0)2.\mathrm{f}(2)=\mathrm{f}(1)-\mathrm{f}(0) 2. f^{\prime \prime}(2)-2 f^{\prime}(1)=12Whichoftheaboveis/arecorrect?Which of the above is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

C. Both 1 and 2

Step-by-step solution

1. f(1)−f(0)=4−6=−2f(2)=8−20+4+6=−2Hence f(2)=f(1)−f(0)\begin{aligned} & f(1)-f(0)=4-6 \\ & =-2 \\ & f(2)=8-20+4+6=-2 \\ & \text{Hence } f(2)=f(1)-f(0)\end{aligned}\thereforeStatement(1)iscorrect.Statement (1) is correct. 2. f′′(2)−2f′(1)=2−2(−5)f′′(2)−2f′(1)=12\begin{aligned} & \text{ 2. } f^{\prime \prime}(2)-2 f^{\prime}(1)=2-2(-5) \\ & f^{\prime \prime}(2)-2 \mathrm{f}^{\prime}(1)=12 \end{aligned}∴\therefore Statement (2) is correct.
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