Q44 / 90 · JEE Main 2024 (04 Apr Shift 1) · Mathematics

Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :

  • A
    2
  • B
    8
  • C
    4
  • Correct answer
    D
    16

Solution

$\begin{aligned} & f(x)=2 \\ & \text { when } x=0 \\ & \because g^{\prime}(f(x)) f^{\prime}(x)=1 \\ & g^{\prime}(2)=\frac{1}{f^{\prime}(0)} \\ & \because f^{\prime}(x)=5 x^4+\frac{2}{4} e^{x / 4} \\ & g^{\prime}(2)=2 \\ & \text { Ans }=16 \\ & \text { Option (1) }\end{aligned}$

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

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