Q238 / 270 · NDA 2022 (Phase 2) · Mathematics

Let $f(x)$ be a function such that $f^{\prime}(x)=g(x)$ and $f^{\prime \prime}(x)$ $=-f(x)$. Let $h(x)=\{f(x)\}^2+\{g(x)\}^2$. Then consider the following statements : 1. $h^{\prime}(3)=0$ 2. $h(1)=h(2)$ Which of the statements given above is/are correct?

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

Solution

$\begin{aligned} & h(x)=\{f(x)\}^2+\{g(x)\}^2 \\ & =\{f(x)\}^2+\left\{f^{\prime}(x)\right\}^2 \\ & h^{\prime}(x)=2 f(x) \cdot f^{\prime}(x)+2 f^{\prime}(x) f^{\prime \prime}(x) \\ & =2 f(x) f^{\prime}(x)-2 f(x) f^{\prime}(x)=0 \\ & \because \quad \text { h' }(x)=0 \\ & \therefore \quad h^{\prime}(3)=0 \end{aligned}$ Statement 1 is correct. $\begin{aligned} & h^{\prime}(x)=0 \Rightarrow h(x)=\text { constant function } \\ & \therefore \quad h(1)=f(2) \end{aligned}$ Statement 2 is correct.

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

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