Q214 / 270 · NDA 2016 (Phase 1) · Mathematics

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

  • A
    1 only
  • B
    2 only
  • Correct answer
    C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

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

Hence, the correct answer is option: 'C'.

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