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Question Number 199033 by essaad last updated on 27/Oct/23

Commented by essaad last updated on 27/Oct/23

$$mrWcouldyoupleasehelpus$$

Commented by mr W last updated on 27/Oct/23

$$whatistodo?$$$$canyourecognisewhatthequestion$$$$is?i{can}^{\prime }t!$$

Commented by mr W last updated on 27/Oct/23

$$btw:thisappwasspeciallydeveloped$$$$foryoutowritenicemathematical$$$$formulas.why{don}^{\prime }tyouuseitfor$$$$writingniceformulasinsteadof$$$$postingbadimages?$$

Answered by witcher3 last updated on 27/Oct/23

$$\mid {\tan }^{-1}\left({\mathrm{x}}^{\frac{1}{\mathrm{n}}}\right)-{\tan }^{-1}\left({\mathrm{y}}^{\frac{1}{\mathrm{n}}}\right)\mid ⩽\frac{1}{2\mathrm{n}}\mid \mathrm{x}-\mathrm{y}\mid$$$$\mathrm{f}\left(\mathrm{t}\right)={\tan }^{-1}\left({\mathrm{t}}^{\frac{1}{\mathrm{n}}}\right),\mathrm{\forall }\mathrm{n}\in {\mathbb{N}}^{\ast }\frac{1}{\mathrm{n}}-1⩽0$$$${\mathrm{f}}^{\prime }\left(\mathrm{t}\right)=\frac{{\mathrm{t}}^{\frac{1}{\mathrm{n}}-1}}{\mathrm{n}}.\frac{1}{(1+{\mathrm{t}}^{\frac{1}{2\mathrm{n}}})},{\mathrm{f}}^{\prime }\left(\mathrm{t}\right)⩽\frac{1}{2\mathrm{n}},\mathrm{\forall }\mathrm{t}⩾1$$$$\mid {\tan }^{-1}\left({\mathrm{x}}^{\frac{1}{\mathrm{n}}}\right)-{\tan }^{-1}\left({\mathrm{y}}^{\frac{1}{\mathrm{n}}}\right)\mid =\mid {\int }_{\mathrm{y}}^{\mathrm{x}}{\mathrm{f}}^{\prime }\left(\mathrm{t}\right)\mathrm{dt}\mid$$$$⩽\mid \frac{1}{2\mathrm{n}}{\int }_{\mathrm{y}}^{\mathrm{x}}1\mathrm{dt}\mid =\frac{1}{2\mathrm{n}}\mid \mathrm{x}-\mathrm{y}\mid$$

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