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inequalities with absolute value

Source: LIMIT 2019 CCS2 P2

April 28, 2021
inequalitiesabsolute value

Problem Statement

Let x,y∈[0,∞)x,y\in[0,\infty). Which of the following is true? <spanclass=′latex−bold′>(A)</span> ∣log⁡(1+x2)−log⁡(1+y2)∣≤∣x−y∣<span class='latex-bold'>(A)</span>~\left|\log\left(1+x^2\right)-\log\left(1+y^2\right)\right|\le|x-y| <spanclass=′latex−bold′>(B)</span> ∣sin⁡2x−sin⁡2y∣≤∣x−y∣<span class='latex-bold'>(B)</span>~\left|\sin^2x-\sin^2y\right|\le|x-y| <spanclass=′latex−bold′>(C)</span> ∣tan⁡−1x−tan⁡−1y∣≤∣x−y∣<span class='latex-bold'>(C)</span>~\left|\tan^{-1}x-\tan^{-1}y\right|\le|x-y| <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
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