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} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\::\:\left[\mathrm{a}\:,\:\mathrm{b}\right]\:\rightarrow\:\left[\mathrm{c}\:,\:\mathrm{d}\right] \\ $$$$\mathrm{a}\:\mathrm{continuous}\:\mathrm{function} \\ $$$$\mathrm{for}\:\mathrm{which}\:\:\exists\lambda\:\in\:\left(\mathrm{a}\:,\:\mathrm{b}\right) \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{a}\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\lambda}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:+\:\mathrm{b}\:\int_{\boldsymbol{\mathrm{b}}} ^{\:\boldsymbol{\lambda}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:\geqslant\:\mathrm{a}\:+\:\mathrm{c} \\…
Question Number 219957 by hardmath last updated on 04/May/25 $$\mathrm{if}\:\:\:\mathrm{f}:\left(\mathrm{0},\infty\right)\rightarrow\left(\mathrm{0},\infty\right) \\ $$$$\:\:\:\:\:\:\mathrm{f}\:\:\mathrm{continuous} \\ $$$$\mathrm{and}\:\:\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{y}}\right)\:=\:\mathrm{2f}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right) \\ $$$$\forall\:\mathrm{x},\mathrm{y}\:>\:\mathrm{0}\:\:\:\mathrm{then}\:\:\:\forall\:\mathrm{a},\mathrm{b}\:>\:\mathrm{0}: \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}+\mathrm{z}}\right)\mathrm{dxdydz}\:=\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \int_{\boldsymbol{\mathrm{a}}}…
Question Number 220022 by hardmath last updated on 04/May/25 $$\mathrm{If}\:\:\:\mathrm{0}\:\leqslant\:\mathrm{a}\:\leqslant\:\mathrm{b}\:\leqslant\:\mathrm{1} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\frac{\mathrm{dxdydz}}{\:\sqrt{\mathrm{1}\:+\:\mathrm{xyz}}}\:\geqslant\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}} }}\: \\…
Question Number 220016 by fantastic last updated on 04/May/25 Answered by efronzo1 last updated on 04/May/25 $$\:\frac{\mathrm{PO}}{\mathrm{sin}\:\mathrm{60}°}\:=\:\frac{\mathrm{8}}{\mathrm{sin}\:\mathrm{75}°}\:=\:\frac{\mathrm{QO}}{\mathrm{sin}\:\mathrm{45}°} \\ $$$$\:\mathrm{PO}\:=\:\frac{\mathrm{16}\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{6}}\:+\sqrt{\mathrm{2}}}\:=\mathrm{4}\left(\mathrm{3}\sqrt{\mathrm{2}}\:−\sqrt{\mathrm{6}}\:\right) \\ $$$$\:\mathrm{area}\:\Delta\mathrm{POQ}\:=\:\frac{\mathrm{1}}{\mathrm{2}}.\:\mathrm{8}.\:\mathrm{4}\sqrt{\mathrm{2}}\:\left(\mathrm{3}−\sqrt{\mathrm{3}}\right) \\ $$$$\:=\:\mathrm{16}\sqrt{\mathrm{2}}\:\left(\mathrm{3}−\sqrt{\mathrm{3}}\:\right) \\ $$…
Question Number 219952 by Mamadi last updated on 04/May/25 $${solve}\:{the}\:{system}\:{differential} \\ $$$${x}'=\mathrm{3}{x}−{y}+{z} \\ $$$${y}'=\mathrm{2}{x}+{z} \\ $$$${z}'=−\mathrm{2}{x}+{y} \\ $$$${with}\:{x},{y},{and}\:{z}\:{are}\:{the}\:{function}\:{of}\: \\ $$$${t}. \\ $$$$ \\ $$ Commented…
Question Number 220019 by fantastic last updated on 04/May/25 $${please}\:{help}\:{me}\:.{The}\:{photos}\:{I}\:{upload} \\ $$$${becomes}\:{blurred} \\ $$$${HELP}\:{please} \\ $$ Commented by Tinku Tara last updated on 05/May/25 $$\mathrm{Please}\:\mathrm{upload}\:\mathrm{smaller}\:\mathrm{pictures}…
Question Number 219948 by Spillover last updated on 04/May/25 Answered by Spillover last updated on 04/May/25 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 219949 by Spillover last updated on 04/May/25 Answered by vnm last updated on 04/May/25 $$\mathrm{Let}\:{AH}=\mathrm{1}\:\mathrm{be}\:\mathrm{the}\:\mathrm{altitude}\:\mathrm{of}\: \\ $$$$\mathrm{triangle}\:{ABC} \\ $$$$\measuredangle{BAH}=\mathrm{60}°,\:\measuredangle{MAH}=\mathrm{45}°−{y} \\ $$$$\measuredangle{CAH}=\mathrm{75}°,\:\measuredangle{NAH}=\mathrm{45}°+{y} \\ $$$$\mathrm{tan}\:\mathrm{60}°−\mathrm{tan}\:\left(\mathrm{45}°−{y}\right)=\mathrm{tan}\:\mathrm{75}°−\mathrm{tan}\:\left(\mathrm{45}°+{y}\right)…