Question Number 208855 by efronzo1 last updated on 25/Jun/24
![](https://www.tinkutara.com/question/34215.png)
Answered by mr W last updated on 25/Jun/24
![let t=1+∣x∣≥1 ((2024^(−t) −λ)/(2024^(−t) −λ^(−1) ))=λ 2024^t 2024^(−t) −λ=λ−2024^t 2λ=2024^(−t) +2024^t =2024^t (1+(1/(2024^(2t) )))>2024 ⇒λ>1012 ⇒∣λ∣_(min) =1013](https://www.tinkutara.com/question/Q208863.png)
$${let}\:{t}=\mathrm{1}+\mid{x}\mid\geqslant\mathrm{1} \\ $$$$\frac{\mathrm{2024}^{−{t}} −\lambda}{\mathrm{2024}^{−{t}} −\lambda^{−\mathrm{1}} }=\lambda\:\mathrm{2024}^{{t}} \\ $$$$\mathrm{2024}^{−{t}} −\lambda=\lambda−\mathrm{2024}^{{t}} \\ $$$$\mathrm{2}\lambda=\mathrm{2024}^{−{t}} +\mathrm{2024}^{{t}} =\mathrm{2024}^{{t}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2024}^{\mathrm{2}{t}} }\right)>\mathrm{2024} \\ $$$$\Rightarrow\lambda>\mathrm{1012} \\ $$$$\Rightarrow\mid\lambda\mid_{{min}} =\mathrm{1013} \\ $$