Applications of Derivatives
Comparison of Functions — Derivative Bounds
nta_pyq_2023_jan
Grade None
Question:
Let $f(x)=2x+\tan^{-1}x$ and $g(x)=\log_e\!\left(\sqrt{1+x^2}+x\right)$, $x\in[0,3]$. Then:
There exists $\hat{x}\in[0,3]$ such that $f'(\hat{x})<g'(\hat{x})$
$\max f(x)>\max g(x)$
There exist $0<x_1<x_2<3$ such that $f(x)<g(x)$ $\forall x\in(x_1,x_2)$
$\min f'(x)=1+\max g'(x)$
Step-by-Step Solution
Key Concept: $f'(x)=2+\frac{1}{1+x^2}\in\left[\frac{21}{10},3\right]$. $g'(x)=\frac{1}{\sqrt{1+x^2}}\in\left[\frac{1}{\sqrt{10}},1\right]$. So $g'(x)<f'(x)$ always — option (1) is false.
$\max f(x)>\max g(x)$.
Correct Answer: 2