Question Number 220097 by Nicholas666 last updated on 05/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\:\in\:\mathbb{Q}\:\:\:;\:\:\:\:{x}\:\neq\:\mathrm{1} \\ $$$$\:\frac{\mathrm{7}}{{x}\:−\:\mathrm{1}}\:+\:\frac{\mathrm{6}}{{x}}\:−\:\frac{\mathrm{4}}{{x}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{3}{x}\:+\:\mathrm{5}}{{x}^{\mathrm{2}} \:−\:\mathrm{1}}\:=\:\frac{\mathrm{1}}{{x}} \\ $$$$\:\:\:\:\:\:\: \\ $$ Answered by Rasheed.Sindhi last updated on…
Question Number 220020 by hardmath last updated on 04/May/25 $$\mathrm{If}\:\:\:\mathrm{f}:\left[\mathrm{a},\mathrm{b}\right]\rightarrow\left[−\mathrm{1},\infty\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{a},\mathrm{b}\in\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}\:\leqslant\:\mathrm{b} \\ $$$$\:\:\:\:\:\:\:\mathrm{f}-\mathrm{continuous} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\left(\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\left(\mathrm{1}+\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}\right)^{\mathrm{3}} \geqslant\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{3}} +\:\mathrm{3}\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \:\int_{\boldsymbol{\mathrm{a}}}…
Question Number 219956 by hardmath last updated on 04/May/25 $$\mathrm{let}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:>\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\::\:\left[\mathrm{a}\:,\:\mathrm{b}\right]\:\rightarrow\:\left[\mathrm{c}\:,\:\mathrm{d}\right] \\ $$$$\mathrm{a}\:\mathrm{continuous}\:\mathrm{function} \\ $$$$\mathrm{for}\:\mathrm{which}\:\:\exists\lambda\:\in\:\left(\mathrm{a}\:,\:\mathrm{b}\right) \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{a}\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\lambda}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:+\:\mathrm{b}\:\int_{\boldsymbol{\mathrm{b}}} ^{\:\boldsymbol{\lambda}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:\geqslant\:\mathrm{a}\:+\:\mathrm{c} \\…
Question Number 220022 by hardmath last updated on 04/May/25 $$\mathrm{If}\:\:\:\mathrm{0}\:\leqslant\:\mathrm{a}\:\leqslant\:\mathrm{b}\:\leqslant\:\mathrm{1} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\frac{\mathrm{dxdydz}}{\:\sqrt{\mathrm{1}\:+\:\mathrm{xyz}}}\:\geqslant\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}} }}\: \\…
Question Number 219957 by hardmath last updated on 04/May/25 $$\mathrm{if}\:\:\:\mathrm{f}:\left(\mathrm{0},\infty\right)\rightarrow\left(\mathrm{0},\infty\right) \\ $$$$\:\:\:\:\:\:\mathrm{f}\:\:\mathrm{continuous} \\ $$$$\mathrm{and}\:\:\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{y}}\right)\:=\:\mathrm{2f}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right) \\ $$$$\forall\:\mathrm{x},\mathrm{y}\:>\:\mathrm{0}\:\:\:\mathrm{then}\:\:\:\forall\:\mathrm{a},\mathrm{b}\:>\:\mathrm{0}: \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}+\mathrm{z}}\right)\mathrm{dxdydz}\:=\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \int_{\boldsymbol{\mathrm{a}}}…
Question Number 220072 by hardmath last updated on 04/May/25 $$\mathrm{If}\:\:\:\mathrm{x},\mathrm{y}\in\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right) \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{log}_{\boldsymbol{\mathrm{sinx}}} ^{\mathrm{2}} \:\left(\frac{\mathrm{sin2x}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\right)\:+\:\mathrm{log}_{\boldsymbol{\mathrm{cosx}}} ^{\mathrm{2}} \:\left(\frac{\mathrm{sin2x}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\right)\:\geqslant\:\mathrm{2} \\ $$ Answered by MrGaster last updated…
Question Number 220069 by hardmath last updated on 04/May/25 $$\mathrm{Let}\:\mathrm{be}\:\:\:\left(\mathrm{H}_{\boldsymbol{\mathrm{n}}} \right)_{\boldsymbol{\mathrm{n}}\geqslant\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{H}_{\boldsymbol{\mathrm{n}}} \:=\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{k}} \\ $$$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{e}^{\mathrm{2}\boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} } \:\left(\sqrt[{\boldsymbol{\mathrm{n}}+\mathrm{1}}]{\left(\mathrm{n}+\mathrm{1}\right)!}\:−\:\sqrt[{\boldsymbol{\mathrm{n}}}]{\mathrm{n}!}\:\right)\:\mathrm{sin}\:\frac{\pi}{\mathrm{n}^{\mathrm{2}} }\:=\:? \\ $$ Answered…
Question Number 219898 by MrGaster last updated on 03/May/25 Answered by Frix last updated on 03/May/25 $$\mathrm{Obviously}\:{x}=\mathrm{8} \\ $$$$\mathrm{No}\:\mathrm{other}\:\mathrm{solution}. \\ $$ Commented by MrGaster last…
Question Number 219899 by MrGaster last updated on 03/May/25 Answered by Frix last updated on 03/May/25 $${y}=\frac{\mathrm{10}−{x}^{\mathrm{3}} }{\mathrm{9}{x}^{\mathrm{2}} } \\ $$$$\Rightarrow \\ $$$${x}^{\mathrm{9}} −\mathrm{201}{x}^{\mathrm{6}} +\mathrm{25}{x}^{\mathrm{3}}…
Question Number 219884 by Rojarani last updated on 03/May/25 $$\:\left({a},{b},{c}\right)>\mathrm{0}\:{such}\:{that}, \\ $$$$\:\:{a}+{b}+{c}=\mathrm{13},\:\:{abc}=\mathrm{36} \\ $$$$\:\:{find}\:{the}\:{maximum}\:{and}\:{minimum}\: \\ $$$$\:{value}\:{of}\:\:{ab}+{bc}+{ca}=? \\ $$ Answered by MrGaster last updated on 03/May/25…