Cho hai số thực x, y thỏa mãn:$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacaWG5b % WaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4naiaadMhacqGHRaWk % caaIYaGaamiEamaakaaabaGaaGymaiabgkHiTiaadIhaaSqabaGccq % GH9aqpcaaIZaWaaOaaaeaacaaIXaGaeyOeI0IaamiEaaWcbeaakiab % gUcaRiaaiodadaqadaqaaiaaikdacaWG5bWaaWbaaSqabeaacaaIYa % aaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaaaa!4C9C! 2{y^3} + 7y + 2x\sqrt {1 - x} = 3\sqrt {1 - x} + 3\left( {2{y^2} + 1} \right)$ . Tìm giá trị lớn nhất của biểu thức P = x + 2y .

$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacaWG5b % WaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4naiaadMhacqGHRaWk % caaIYaGaamiEamaakaaabaGaaGymaiabgkHiTiaadIhaaSqabaGccq % GH9aqpcaaIZaWaaOaaaeaacaaIXaGaeyOeI0IaamiEaaWcbeaakiab % gUcaRiaaiodadaqadaqaaiaaikdacaWG5bWaaWbaaSqabeaacaaIYa % aaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaaaa!4C9C! 2{y^3} + 7y + 2x\sqrt {1 - x} = 3\sqrt {1 - x} + 3\left( {2{y^2} + 1} \right)$

$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgsDiBlaaik % dadaqadaqaaiaadMhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaI % ZaGaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiodacaWG5b % GaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaamyE % aiabgkHiTiaaigdaaiaawIcacaGLPaaacqGH9aqpcaaIYaWaaeWaae % aacaaIXaGaeyOeI0IaamiEaaGaayjkaiaawMcaamaakaaabaGaaGym % aiabgkHiTiaadIhaaSqabaGccqGHRaWkcaaIZaWaaOaaaeaacaaIXa % GaeyOeI0IaamiEaaWcbeaakiabgkHiTiaaikdadaGcaaqaaiaaigda % cqGHsislcaWG4baaleqaaaaa!5AF9! \Leftrightarrow 2\left( {{y^3} - 3{y^2} + 3y - 1} \right) + \left( {y - 1} \right) = 2\left( {1 - x} \right)\sqrt {1 - x} + 3\sqrt {1 - x} - 2\sqrt {1 - x}$

$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgsDiBlaaik % dadaqadaqaaiaadMhacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWba % aSqabeaacaaIZaaaaOGaey4kaSYaaeWaaeaacaWG5bGaeyOeI0IaaG % ymaaGaayjkaiaawMcaaiabg2da9iaaikdadaqadaqaamaakaaabaGa % aGymaiabgkHiTiaadIhaaSqabaaakiaawIcacaGLPaaadaahaaWcbe % qaaiaaiodaaaGccqGHRaWkdaGcaaqaaiaaigdacqGHsislcaWG4baa % leqaaOGaaGPaVlaaykW7daqadaqaaiaaigdaaiaawIcacaGLPaaaaa % a!5344! \Leftrightarrow 2{\left( {y - 1} \right)^3} + \left( {y - 1} \right) = 2{\left( {\sqrt {1 - x} } \right)^3} + \sqrt {1 - x} \,\,\left( 1 \right)$

Xét hàm số $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqada % qaaiaadshaaiaawIcacaGLPaaacqGH9aqpcaaIYaGaamiDamaaCaaa % leqabaGaaG4maaaakiabgUcaRiaadshaaaa!3EE0! f\left( t \right) = 2{t^3} + t$ trên $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaajibabaGaaG % imaiaacUdacaaMc8Uaey4kaSIaeyOhIukacaGLBbGaayzkaaaaaa!3D13! \left[ {0;\, + \infty } \right)$.

Ta có: $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadAgagaqbam % aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaiAdacaWG0bWa % aWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaaaa!3EB1! f'\left( t \right) = 6{t^2} + 1 > 0$ với $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgcGiIiaads % hacqGHLjYScaaIWaaaaa!3A32! \forall t \ge 0$ $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkDiElaadA % gadaqadaqaaiaadshaaiaawIcacaGLPaaaaaa!3BB3! \Rightarrow f\left( t \right)$ luôn đồng biến trên $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaajibabaGaaG % imaiaacUdacaaMc8Uaey4kaSIaeyOhIukacaGLBbGaayzkaaaaaa!3D13! \left[ {0;\, + \infty } \right)$.

Vậy $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG % ymaaGaayjkaiaawMcaaiabgsDiBlaadMhacqGHsislcaaIXaGaeyyp % a0ZaaOaaaeaacaaIXaGaeyOeI0IaamiEaaWcbeaaaaa!40F5! \left( 1 \right) \Leftrightarrow y - 1 = \sqrt {1 - x}$$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgsDiBlaadM % hacqGH9aqpcaaIXaGaey4kaSYaaOaaaeaacaaIXaGaeyOeI0IaamiE % aaWcbeaaaaa!3EA6! \Leftrightarrow y = 1 + \sqrt {1 - x}$.

$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkDiElaadc % facqGH9aqpcaWG4bGaey4kaSIaaGOmaiaadMhacqGH9aqpcaWG4bGa % ey4kaSIaaGOmaiabgUcaRiaaikdadaGcaaqaaiaaigdacqGHsislca % WG4baaleqaaaaa!45B9! \Rightarrow P = x + 2y = x + 2 + 2\sqrt {1 - x}$ với  $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam % iEaiabgsMiJkaaigdaaiaawIcacaGLPaaaaaa!3ADF! \left( {x \le 1} \right)$.

Xét hàm số $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaqada % qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaIYaGaey4kaSIaamiE % aiabgUcaRiaaikdadaGcaaqaaiaaigdacqGHsislcaWG4baaleqaaa % aa!415A! g\left( x \right) = 2 + x + 2\sqrt {1 - x}$ trên $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaajadabaGaey % OeI0IaeyOhIuQaai4oaiaaykW7caaIXaaacaGLOaGaayzxaaaaaa!3D3E! \left( { - \infty ;\,1} \right]$.

Ta có: $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadEgagaqbam % aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl % daWcaaqaaiaaigdaaeaadaGcaaqaaiaaigdacqGHsislcaWG4baale % qaaaaaaaa!3FA0! g'\left( x \right) = 1 - \frac{1}{{\sqrt {1 - x} }}$$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabg2da9maala % aabaWaaOaaaeaacaaIXaGaeyOeI0IaamiEaaWcbeaakiabgkHiTiaa % igdaaeaadaGcaaqaaiaaigdacqGHsislcaWG4baaleqaaaaaaaa!3E31! = \frac{{\sqrt {1 - x} - 1}}{{\sqrt {1 - x} }}$ .$% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadEgagaqbam % aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaaicdacqGHshI3 % caWG4bGaeyypa0JaaGimaaaa!4041! g'\left( x \right) = 0 \Rightarrow x = 0$

Bảng biến thiên g(x):

Từ bảng biến thiên của hàm số g(x) suy ra giá trị lớn nhất của P là: $% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxababaGaci % yBaiaacggacaGG4baaleaadaqcWaqaaiabgkHiTiabg6HiLkaacUda % caaMc8UaaGymaaGaayjkaiaaw2faaaqabaGccaWGNbWaaeWaaeaaca % WG4baacaGLOaGaayzkaaGaeyypa0JaaGinaaaa!458B! \mathop {\max }\limits_{\left( { - \infty ;\,1} \right]} g\left( x \right) = 4$