Mathematics · Methods of Differentiation

JEE Main 2024 — 5 April, Shift 1 — Question 17

Let $f(x)=x^{5}+2 x^{3}+3 x+1, x \in R$ , and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in R$ . Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :

Option A:
7
Option B:
42
Option C:
1
Option D:
14
Correct

Answer: D

Step-by-step solution

$f(x)=x^{5}+2 x^{3}+3 x+1$

$f^{\prime}(x)=5 \mathrm{x}^{4}+6 \mathrm{x}^{2}+3$

$\mathrm{f}^{\prime}(1)=5+6+3=14$

$\mathrm{g}(\mathrm{f}(\mathrm{x}))=\mathrm{x}$

$\mathrm{g}^{\prime}(\mathrm{f}(\mathrm{x})) \mathrm{f}^{\prime}(\mathrm{x})=1$

for $f(x)=7$

$\Rightarrow \mathrm{x}^{5}+2 \mathrm{x}^{3}+3 \mathrm{x}+1=7$

$\Rightarrow \mathrm{x}=1$

$g^{\prime}(7) f^{\prime}(1)=1 \Rightarrow g^{\prime}(7)=\frac{1}{f^{\prime}(1)}=\frac{1}{14}$

$\mathrm{x}=1, \mathrm{f}(\mathrm{x})=7 \Rightarrow \mathrm{~g}(7)=1$

$\frac{g(7)}{g^{\prime}(7)}=\frac{1}{1 / 14}=14$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 5 April, Shift 1
Subject: Mathematics
Chapter: Methods of Differentiation
Topic: First, second, third derivative of inverse of a function