Tìm trên mỗi nhánh của đồ thị (C): \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacqGH9a % qpdaWcaaqaaiaaisdacaWG4bGaeyOeI0IaaGyoaaqaaiaadIhacqGH % sislcaaIZaaaaaaa!3E0F! y = \frac{{4x - 9}}{{x - 3}}\) các điểm \(M_1;M_2\)  để độ dài \(M_1M_2\) đạt giá trị nhỏ nhất, giá trị nhỏ nhất đó bằng:

Lấy \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa % aaleaacaaIXaaabeaakmaabmaabaGaamiEamaaBaaaleaacaaIXaaa % beaakiabgUcaRiaaiodacaGG7aGaaGPaVlaaisdacqGHRaWkdaWcaa % qaaiaaiodaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjk % aiaawMcaaaaa!4372! {M_1}\left( {{x_1} + 3;\,4 + \frac{3}{{{x_1}}}} \right)\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa % aaleaacaaIXaaabeaakiabg6da+iaaicdaaaa!39A4! ;{x_1} > 0\); \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa % aaleaacaaIYaaabeaakmaabmaabaGaamiEamaaBaaaleaacaaIYaaa % beaakiabgUcaRiaaiodacaGG7aGaaGPaVlaaisdacqGHRaWkdaWcaa % qaaiaaiodaaeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjk % aiaawMcaaaaa!4375! {M_2}\left( {{x_2} + 3;\,4 + \frac{3}{{{x_2}}}} \right)\); \(x_2 < 0\)

Khi đó \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa % aaleaacaaIXaaabeaakiaad2eadaWgaaWcbaGaaGOmaaqabaGcdaah % aaWcbeqaaiaaikdaaaGccqGH9aqpdaqadaqaaiaadIhadaWgaaWcba % GaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaaikdaaeqaaaGc % caGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXa % Gaey4kaSYaaSaaaeaacaaI5aaabaGaamiEamaaDaaaleaacaaIXaaa % baGaaGOmaaaakiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaa % GccaGLOaGaayzkaaaaaa!4C0A! {M_1}{M_2}^2 = {\left( {{x_1} - {x_2}} \right)^2}\left( {1 + \frac{9}{{x_1^2x_2^2}}} \right)\)

Áp dụng bất đẳng thức Cô Si ta có \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca % WG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaa % caaIYaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaki % abgwMiZkaaisdadaabdaqaaiaadIhadaWgaaWcbaGaaGymaaqabaGc % caWG4bWaaSbaaSqaaiaaikdaaeqaaaGccaGLhWUaayjcSdaaaa!46BD! {\left( {{x_1} - {x_2}} \right)^2} \ge 4\left| {{x_1}{x_2}} \right|\) và \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU % caRmaalaaabaGaaGyoaaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaa % ikdaaaGccaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiabgw % MiZoaalaaabaGaaGOnaaqaamaaemaabaGaamiEamaaBaaaleaacaaI % XaaabeaakiaadIhadaWgaaWcbaGaaGOmaaqabaaakiaawEa7caGLiW % oaaaaaaa!4750! 1 + \frac{9}{{x_1^2x_2^2}} \ge \frac{6}{{\left| {{x_1}{x_2}} \right|}}\).

Suy ra \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa % aaleaacaaIXaaabeaakiaad2eadaWgaaWcbaGaaGOmaaqabaGcdaah % aaWcbeqaaiaaikdaaaGccqGHLjYScaaIYaGaaGinaiabgkDiElaad2 % eadaWgaaWcbaGaaGymaaqabaGccaWGnbWaaSbaaSqaaiaaikdaaeqa % aOGaeyyzImRaaGOmamaakaaabaGaaGOnaaWcbeaaaaa!46EF! {M_1}{M_2}^2 \ge 24 \Rightarrow {M_1}{M_2} \ge 2\sqrt 6 \)

Độ dài \(M_1M_2\) đạt giá trị nhỏ nhất bẳng \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaka % aabaGaaGOnaaWcbeaaaaa!378B! 2\sqrt 6 \) khi  \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqGHsislcaWG % 4bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamiEamaaDaaaleaacaaIXa % aabaGaaGinaaaakiabg2da9iaaiMdaaaGaay5Eaaaaaa!415B! \left\{ \begin{array}{l} {x_1} = - {x_2}\\ x_1^4 = 9 \end{array} \right.\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aai % qaaqaabeqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGH9aqpdaGc % aaqaaiaaiodaaSqabaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO % Gaeyypa0JaeyOeI0YaaOaaaeaacaaIZaaaleqaaaaakiaawUhaaaaa % !420B! \Leftrightarrow \left\{ \begin{array}{l} {x_1} = \sqrt 3 \\ {x_2} = - \sqrt 3 \end{array} \right.\)