Q96 / 120 · NDA 2024 (Phase 2) · Mathematics

Paragraph: Let $f:(-1,1) \rightarrow \mathrm{R}$ be a differentiable function with $f(0)=-1$ and $f^{\prime}(0)=1$. Let $h(x)=f(2 f(x)+2)$ and $g(x)=(h(x))^2$.
Question: What is $g^{\prime}(0)$ equal to?

  • Correct answer
    A
    -4
  • B
    -2
  • C
    0
  • D
    4

Solution

Let $\mathrm{f}:(-1,1) \rightarrow \mathrm{R}$ be a differentiable function with $\mathrm{f}(0)$ $=-1$ and $f^{\prime}(0)=1$ $\mathrm{g}(\mathrm{n})=(\mathrm{h}(\mathrm{x}))^2$ Differentiate both side w.r.t $x$ $\mathrm{g}^{\prime}(\mathrm{x})=2(\mathrm{~h}(\mathrm{x})) \cdot \mathrm{h}^{\prime}(\mathrm{x})$ $\begin{aligned} & g^{\prime}(0)=2(h(0)) \cdot h^{\prime}(0) \\ & {\left[\because h^{\prime}(0)=2\right]} \\ & {\left[\because f^{\prime}(0)=-1\right]} \\ & =2 f(2+10)+2) \cdot 2 \\ & =2 f(2(-1)+2)=2 f(0) \cdot 2=(-2) \cdot 2=-4 .\end{aligned}$

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

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