Tất cả các giá trị thực của m để bất phương trình \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaka % aabaGaamiEaaWcbeaakiabgUcaRmaakaaabaGaamiEaiabgUcaRiaa % igdacaaIYaaaleqaaOGaeyizImQaamyBaiaac6caciGGSbGaai4Bai % aacEgadaWgaaWcbaGaaGynaiabgkHiTmaakaaabaGaaGinaiabgkHi % TiaadIhaaWqabaaaleqaaOGaaG4maaaa!4807! x\sqrt x + \sqrt {x + 12} \le m.{\log _{5 - \sqrt {4 - x} }}3\) có nghiệm là

Điều kiện: \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI % GiopaadmaabaGaaGimaiaacUdacaaI0aaacaGLBbGaayzxaaaaaa!3C9E! x \in \left[ {0;4} \right]\) . Ta thấy \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgk % HiTiaadIhacqGHKjYOcaaI0aGaeyO0H4TaaGynaiabgkHiTmaakaaa % baGaaGinaiabgkHiTiaadIhaaSqabaGccqGHLjYScaaIZaGaeyO0H4 % TaciiBaiaac+gacaGGNbWaaSbaaSqaaiaaiwdacqGHsisldaGcaaqa % aiaaisdacqGHsislcaWG4baameqaaaWcbeaakiaaiodacqGH+aGpca % aIWaaaaa!50C5! 4 - x \le 4 \Rightarrow 5 - \sqrt {4 - x} \ge 3 \Rightarrow {\log _{5 - \sqrt {4 - x} }}3 > 0\)

Khi đó bất phương trình đã cho trở thành \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabgw % MiZkaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaqa % daqaaiaadIhadaGcaaqaaiaadIhaaSqabaGccqGHRaWkdaGcaaqaai % aadIhacqGHRaWkcaaIXaGaaGOmaaWcbeaaaOGaayjkaiaawMcaaiaa % c6caciGGSbGaai4BaiaacEgadaWgaaWcbaGaaG4maaqabaGcdaqada % qaaiaaiwdacqGHsisldaGcaaqaaiaaisdacqGHsislcaWG4baaleqa % aaGccaGLOaGaayzkaaaaaa!4F9F! m \ge f\left( x \right) = \left( {x\sqrt x + \sqrt {x + 12} } \right).{\log _3}\left( {5 - \sqrt {4 - x} } \right)(*)\)

Với \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2 % da9iaadIhadaGcaaqaaiaadIhaaSqabaGccqGHRaWkdaGcaaqaaiaa % dIhacqGHRaWkcaaIXaGaaGOmaaWcbeaakiabgkDiElqadwhagaqbai % abg2da9maalaaabaGaaG4mamaakaaabaGaamiEaaWcbeaaaOqaaiaa % ikdaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmamaakaaabaGaam % iEaiabgUcaRiaaigdacaaIYaaaleqaaaaaaaa!4B5E! u = x\sqrt x + \sqrt {x + 12} \Rightarrow u' = \frac{{3\sqrt x }}{2} + \frac{1}{{2\sqrt {x + 12} }}\) và \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2 % da9iGacYgacaGGVbGaai4zamaaBaaaleaacaaIZaaabeaakmaabmaa % baGaaGynaiabgkHiTmaakaaabaGaaGinaiabgkHiTiaadIhaaSqaba % aakiaawIcacaGLPaaacqGHshI3ceWG2bGbauaacqGH9aqpdaWcaaqa % aiaaigdaaeaacaaIYaWaaOaaaeaacaaI0aGaeyOeI0IaamiEaaWcbe % aakmaabmaabaGaaGynaiabgkHiTmaakaaabaGaaGinaiabgkHiTiaa % dIhaaSqabaaakiaawIcacaGLPaaacaGGUaGaciiBaiaac6gacaaIZa % aaaaaa!53CD! v = {\log _3}\left( {5 - \sqrt {4 - x} } \right) \Rightarrow v' = \frac{1}{{2\sqrt {4 - x} \left( {5 - \sqrt {4 - x} } \right).\ln 3}}\)

Suy ra \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafa % WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyOpa4JaaGimaiaacUda % cqGHaiIicaWG4bGaeyicI48aaeWaaeaacaaIWaGaai4oaiaaisdaai % aawIcacaGLPaaacqGHshI3caWGMbWaaeWaaeaacaWG4baacaGLOaGa % ayzkaaaaaa!48D1! f'\left( x \right) > 0;\forall x \in \left( {0;4} \right) \Rightarrow f\left( x \right)\) là hàm số đồng biến trên đoạn [ 0;4]

Để bất phương trình (*) có nghiệm \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaam % yBaiabgwMiZoaaxababaGaciyBaiaacMgacaGGUbaaleaadaWadaqa % aiaaicdacaGG7aGaaGinaaGaay5waiaaw2faaaqabaGccaWGMbWaae % WaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaamOzamaabmaabaGa % aGimaaGaayjkaiaawMcaaiabg2da9iaaikdadaGcaaqaaiaaiodaaS % qabaaaaa!4C85! \Leftrightarrow m \ge \mathop {\min }\limits_{\left[ {0;4} \right]} f\left( x \right) = f\left( 0 \right) = 2\sqrt 3 \)