JEE Main 2022 · 25 Jun · Shift 1 · Mathematics Q13

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MathematicsSingle correct+4 / -1Question 13 of 90

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be two functions defined by $f(x)=\log_e\left(x^2+1\right)-e^{-x}+1$ and $g(x)=\dfrac{1-2e^{2x}}{e^x}$ . Then, for which of the following range of $\alpha$ , the inequality
$f\left(g\left(\frac{(\alpha-1)^2}3\right)\right)>f\left(g\left(\alpha-\frac53\right)\right)$
holds ?

(2,3)

(-2,-1)

(1,2)

(-1,1)

Answer and worked solution

Official answer: A

$f$ is increasing. $f'(x)=\dfrac{2x}{x^2+1}+e^{-x}$ ; for $x\ge0$ both terms are $\ge0$ , and for $x<0$ , $e^{-x}>1\ge\left|\dfrac{2x}{x^2+1}\right|$ .

$g$ is decreasing. $g(x)=e^{-x}-2e^x$ .

So $f\circ g$ is decreasing, and the inequality means $\dfrac{(\alpha-1)^2}3<\alpha-\dfrac53$ , i.e. $\alpha^2-5\alpha+6<0$ : $\alpha\in(2,3)$ .

(A)

Topics:Monotonicity Composition and inverse functions Quadratic expressions

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JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 13