Cho hàm số \(y = f (x)\) có đạo hàm và liên tục trên R . Biết rằng đồ thị hàm số \(y = f' (x)\) như hình dưới đây.
Lập hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadAgadaqadaqaaiaa % dIhaaiaawIcacaGLPaaacqGHsislcaWG4bWaaWbaaSqabeaacaaIYa % aaaOGaeyOeI0IaamiEaaaa!42A4! g\left( x \right) = f\left( x \right) - {x^2} - x\). Mệnh đề nào sau đây đúng?
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Lời giải:
Báo saiXét hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9iqadAgagaqbamaabmaa % baGaamiEaaGaayjkaiaawMcaaiabgkHiTmaabmaabaGaaGOmaiaadI % hacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaa!43B6! h\left( x \right) = f'\left( x \right) - \left( {2x + 1} \right)\). Khi đó hàm số \(h(x)\) liên tục trên các đoạn \([-1;1] ;[1;2]\) , và có \(g(x)\) là một nguyên hàm của hàm số \(y = h(x)\)
Do đó diện tích hình phẳng giới hạn bởi \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadIhacqGH9aqpcqGHsislcaaIXaaabaGaamiEaiabg2da9iaa % igdaaeaacaWG5bGaeyypa0JabmOzayaafaWaaeWaaeaacaWG4baaca % GLOaGaayzkaaaabaGaamyEaiabg2da9iaaikdacaWG4bGaey4kaSIa % aGymaaaacaGL7baaaaa!485B! \left\{ \begin{array}{l} x = - 1\\ x = 1\\ y = f'\left( x \right)\\ y = 2x + 1 \end{array} \right.\) là
\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa % aaleaacaaIXaaabeaakiabg2da9maapehabaWaaqWaaeaaceWGMbGb % auaadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGHsisldaqadaqaai % aaikdacaWG4bGaey4kaSIaaGymaaGaayjkaiaawMcaaaGaay5bSlaa % wIa7aiaabsgacaWG4baaleaacqGHsislcaaIXaaabaGaaGymaaqdcq % GHRiI8aaaa!4BDE! {S_1} = \int\limits_{ - 1}^1 {\left| {f'\left( x \right) - \left( {2x + 1} \right)} \right|{\rm{d}}x} \)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaa8 % qCaeaadaWadaqaaiqadAgagaqbamaabmaabaGaamiEaaGaayjkaiaa % wMcaaiabgkHiTmaabmaabaGaaGOmaiaadIhacqGHRaWkcaaIXaaaca % GLOaGaayzkaaaacaGLBbGaayzxaaGaaeizaiaadIhaaSqaaiabgkHi % TiaaigdaaeaacaaIXaaaniabgUIiYdaaaa!48E5! = \int\limits_{ - 1}^1 {\left[ {f'\left( x \right) - \left( {2x + 1} \right)} \right]{\rm{d}}x} \)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaaq % GaaeaacaWGNbWaaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLiWoa % daqhaaWcbaGaeyOeI0IaaGymaaqaaiaaigdaaaaaaa!3E92! = \left. {g\left( x \right)} \right|_{ - 1}^1=g(1)-g(-1)\)
Vì \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa % aaleaacaaIXaaabeaakiabg6da+iaaicdaaaa!397F! {S_1} > 0\) nên \(g(1)>g(-1)\).
Diện tích hình phẳng giới hạn bởi \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadIhacqGH9aqpcaaIXaaabaGaamiEaiabg2da9iaaikdaaeaa % caWG5bGaeyypa0JabmOzayaafaWaaeWaaeaacaWG4baacaGLOaGaay % zkaaaabaGaamyEaiabg2da9iaaikdacaWG4bGaey4kaSIaaGymaaaa % caGL7baaaaa!476F! \left\{ \begin{array}{l} x = 1\\ x = 2\\ y = f'\left( x \right)\\ y = 2x + 1 \end{array} \right.\) là
\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa % aaleaacaaIYaaabeaakiabg2da9maapehabaWaaqWaaeaaceWGMbGb % auaadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGHsisldaqadaqaai % aaikdacaWG4bGaey4kaSIaaGymaaGaayjkaiaawMcaaaGaay5bSlaa % wIa7aiaabsgacaWG4baaleaacaaIXaaabaGaaGOmaaqdcqGHRiI8aa % aa!4AF3! {S_2} = \int\limits_1^2 {\left| {f'\left( x \right) - \left( {2x + 1} \right)} \right|{\rm{d}}x} \)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaa8 % qCaeaadaWadaqaamaabmaabaGaaGOmaiaadIhacqGHRaWkcaaIXaaa % caGLOaGaayzkaaGaeyOeI0IabmOzayaafaWaaeWaaeaacaWG4baaca % GLOaGaayzkaaaacaGLBbGaayzxaaGaaeizaiaadIhaaSqaaiaaigda % aeaacaaIYaaaniabgUIiYdaaaa!47F9! = \int\limits_1^2 {\left[ {\left( {2x + 1} \right) - f'\left( x \right)} \right]{\rm{d}}x} \)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey % OeI0YaaqGaaeaacaWGNbWaaeWaaeaacaWG4baacaGLOaGaayzkaaaa % caGLiWoadaqhaaWcbaGaaGymaaqaaiaaikdaaaaaaa!3E93! = - \left. {g\left( x \right)} \right|_1^2=g(1) - g(2)\)
Vì \(S_{2}>0\) nên g(1) > g (2)
Đề thi thử tốt nghiệp THPT QG môn Toán năm 2020
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