Cho hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWG5bGaeyypa0ZaaSaaaeaacaWG4bGaey4kaSIaaGymaaqaaiaa % dIhacqGHsislcaaIYaaaaiaaywW7caGGOaGaam4qaiaacMcaaaa!4116! y = \frac{{x + 1}}{{x - 2}}\quad (C)\) . Gọi d là khoảng cách từ giao điểm của hai đường tiệm cận của đồ thị đến một tiếp tuyến của (C). Giá trị lớn nhất mà d có thể đạt được là:

Ta có: \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacE % cadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiab % gkHiTiaaiodaaeaadaqadaqaaiaadIhacqGHsislcaaIYaaacaGLOa % GaayzkaaWaaWbaaSqabeaacaaIYaaaaaaakiaaywW7cqGHaiIicaWG % 4bGaeyiyIKRaaGOmaaaa!47E2! y'\left( x \right) = \frac{{ - 3}}{{{{\left( {x - 2} \right)}^2}}}\quad \forall x \ne 2\). Gọi I là giao của hai tiệm cận nên I(2;1)

Gọi \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaabm % aabaGaamiEamaaBaaaleaacaaIWaaabeaakiaacUdacaWG5bWaaSba % aSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaGaeyypa0Jaamytamaabm % aabaGaamiEamaaBaaaleaacaaIWaaabeaakiaacUdadaWcaaqaaiaa % dIhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaaIXaaabaGaamiEam % aaBaaaleaacaaIWaaabeaakiabgkHiTiaaikdaaaaacaGLOaGaayzk % aaGaeyicI48aaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaa!4DFA! M\left( {{x_0};{y_0}} \right) = M\left( {{x_0};\frac{{{x_0} + 1}}{{{x_0} - 2}}} \right) \in \left( C \right)\)

Khi đó tiếp tuyến tại \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaabm % aabaGaamiEamaaBaaaleaacaaIWaaabeaakiaacUdacaWG5bWaaSba % aSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaaaaa!3CE8! M\left( {{x_0};{y_0}} \right)\) có phương trình:

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaai % OoaiaaysW7caWG5bGaeyypa0JaamyEaiaacEcadaqadaqaaiaadIha % daWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaadaqadaqaaiaadI % hacqGHsislcaWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzk % aaGaey4kaSIaamyEamaaBaaaleaacaaIWaaabeaaaaa!48EE! \Delta :\;y = y'\left( {{x_0}} \right)\left( {x - {x_0}} \right) + {y_0}\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaam % yEaiabg2da9maalaaabaGaeyOeI0IaaG4maaqaamaabmaabaGaamiE % amaaBaaaleaacaaIWaaabeaakiabgkHiTiaaikdaaiaawIcacaGLPa % aadaahaaWcbeqaaiaaikdaaaaaaOWaaeWaaeaacaWG4bGaeyOeI0Ia % amiEamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiabgUcaRm % aalaaabaGaamiEamaaBaaaleaacaaIWaaabeaakiabgUcaRiaaigda % aeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaaGOmaaaaaa % a!4F91! \Leftrightarrow y = \frac{{ - 3}}{{{{\left( {{x_0} - 2} \right)}^2}}}\left( {x - {x_0}} \right) + \frac{{{x_0} + 1}}{{{x_0} - 2}}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aaS % aaaeaacqGHsislcaaIZaaabaWaaeWaaeaacaWG4bWaaSbaaSqaaiaa % icdaaeqaaOGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaaCaaaleqaba % GaaGOmaaaaaaGccaGGUaGaamiEaiabgkHiTiaadMhacqGHRaWkdaWc % aaqaaiaaiodacaWG4bWaaSbaaSqaaiaaicdaaeqaaaGcbaWaaeWaae % aacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaaGOmaaGaayjk % aiaawMcaamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaai % aadIhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaaIXaaabaGaamiE % amaaBaaaleaacaaIWaaabeaakiabgkHiTiaaikdaaaGaeyypa0JaaG % imaaaa!5735! \Leftrightarrow \frac{{ - 3}}{{{{\left( {{x_0} - 2} \right)}^2}}}.x - y + \frac{{3{x_0}}}{{{{\left( {{x_0} - 2} \right)}^2}}} + \frac{{{x_0} + 1}}{{{x_0} - 2}} = 0\)

Khi đó ta có:\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaabm % aabaGaamysaiaacUdacqGHuoaraiaawIcacaGLPaaacqGH9aqpdaWc % aaqaamaaemaabaWaaSaaaeaacqGHsislcaaI2aaabaWaaeWaaeaaca % WG4bWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaaGOmaaGaayjkaiaa % wMcaamaaCaaaleqabaGaaGOmaaaaaaGccqGHsislcaaIXaGaey4kaS % YaaSaaaeaacaaIZaGaamiEamaaBaaaleaacaaIWaaabeaaaOqaamaa % bmaabaGaamiEamaaBaaaleaacaaIWaaabeaakiabgkHiTiaaikdaai % aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaa % aeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaaGymaaqaai % aadIhadaWgaaWcbaGaaGimaaqabaGccqGHsislcaaIYaaaaaGaay5b % SlaawIa7aaqaamaakaaabaGaaGymaiabgUcaRmaalaaabaGaaGyoaa % qaamaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaakiabgkHiTiaa % ikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaaaaaqabaaaaa % aa!6352! d\left( {I;\Delta } \right) = \frac{{\left| {\frac{{ - 6}}{{{{\left( {{x_0} - 2} \right)}^2}}} - 1 + \frac{{3{x_0}}}{{{{\left( {{x_0} - 2} \right)}^2}}} + \frac{{{x_0} + 1}}{{{x_0} - 2}}} \right|}}{{\sqrt {1 + \frac{9}{{{{\left( {{x_0} - 2} \right)}^4}}}} }}\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaam % izamaabmaabaGaamysaiaacUdacqGHuoaraiaawIcacaGLPaaacqGH % 9aqpdaWcaaqaamaaemaabaGaaGOnaiaadIhadaWgaaWcbaGaaGimaa % qabaGccqGHsislcaaIXaGaaGOmaaGaay5bSlaawIa7aaqaamaakaaa % baWaaeWaaeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaaG % OmaaGaayjkaiaawMcaamaaCaaaleqabaGaaGinaaaakiabgUcaRiaa % iMdaaSqabaaaaaaa!4ED2! \Leftrightarrow d\left( {I;\Delta } \right) = \frac{{\left| {6{x_0} - 12} \right|}}{{\sqrt {{{\left( {{x_0} - 2} \right)}^4} + 9} }}\)

Áp dụng BĐT: \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa % aaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaikda % aaGccqGHLjYScaaIYaGaamyyaiaadkgacaaMf8UaeyiaIiIaamyyai % aacYcacaWGIbaaaa!43B2! {a^2} + {b^2} \ge 2ab\quad \forall a,b\)

Tacó: \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiabgU % caRmaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaakiabgkHiTiaa % ikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaGccqGHLjYSca % aIYaGaaiOlaiaaiodacaGGUaWaaeWaaeaacaWG4bWaaSbaaSqaaiaa % icdaaeqaaOGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaaCaaaleqaba % GaaGOmaaaakiabgsDiBpaakaaabaGaaGyoaiabgUcaRmaabmaabaGa % amiEamaaBaaaleaacaaIWaaabeaakiabgkHiTiaaikdaaiaawIcaca % GLPaaadaahaaWcbeqaaiaaisdaaaaabeaakiabgwMiZoaakaaabaGa % aGOnamaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaakiabgkHiTi % aaikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaabeaaaaa!5B24! 9 + {\left( {{x_0} - 2} \right)^4} \ge 2.3.{\left( {{x_0} - 2} \right)^2} \Leftrightarrow \sqrt {9 + {{\left( {{x_0} - 2} \right)}^4}} \ge \sqrt {6{{\left( {{x_0} - 2} \right)}^2}} \)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taam % izamaabmaabaGaamysaiaacUdacqGHuoaraiaawIcacaGLPaaacqGH % 9aqpdaWcaaqaamaaemaabaGaaGOnaiaadIhadaWgaaWcbaGaaGimaa % qabaGccqGHsislcaaIXaGaaGOmaaGaay5bSlaawIa7aaqaamaakaaa % baWaaeWaaeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaaG % OmaaGaayjkaiaawMcaamaaCaaaleqabaGaaGinaaaakiabgUcaRiaa % iMdaaSqabaaaaOGaeyizIm6aaSaaaeaadaabdaqaaiaaiAdacaWG4b % WaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaaGymaiaaikdaaiaawEa7 % caGLiWoaaeaadaGcaaqaaiaaiAdadaqadaqaaiaadIhadaWgaaWcba % GaaGimaaqabaGccqGHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqa % beaacaaIYaaaaaqabaaaaOGaeyypa0ZaaOaaaeaacaaI2aaaleqaaa % aa!6198! \Rightarrow d\left( {I;\Delta } \right) = \frac{{\left| {6{x_0} - 12} \right|}}{{\sqrt {{{\left( {{x_0} - 2} \right)}^4} + 9} }} \le \frac{{\left| {6{x_0} - 12} \right|}}{{\sqrt {6{{\left( {{x_0} - 2} \right)}^2}} }} = \sqrt 6 \)

Vậy giá trị lớn nhất mà  có thể đạt được là: \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca % aI2aaaleqaaaaa!36CE! \sqrt 6\)