Có bao nhiêu điểm thuộc đồ thị hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2 % da9iabgkHiTiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaI % ZaGaamiEaiabgUcaRiaaiodaaaa!3F1C! y = - {x^3} - 3x + 3\) cách giao điểm của đồ thị hàm số với trục tung một khoảng bằng \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca % aIXaGaaG4naaWcbeaaaaa!378B! \sqrt {17} \)?
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Lời giải:
Báo saiGọi A là giao điểm của đồ thị hàm số với trục tung suy ra A(0;3).
Gọi \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaabm % aabaGaamiEamaaBaaaleaacaaIWaaabeaakiaacYcacqGHsislcaWG % 4bWaaSbaaSqaaiaaicdaaeqaaOWaaWbaaSqabeaacaaIZaaaaOGaey % OeI0IaaG4maiaadIhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaaI % ZaaacaGLOaGaayzkaaaaaa!43F0! M\left( {{x_0}, - {x_0}^3 - 3{x_0} + 3} \right)\) là cần tìm. Ta có:
\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaad2 % eacqGH9aqpdaGcaaqaaiaaigdacaaI3aaaleqaaOGaeyi1HSTaamyq % aiaad2eadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIXaGaaG4nai % abgsDiBlaadIhadaqhaaWcbaGaaGimaaqaaiaaikdaaaGccqGHRaWk % daqadaqaaiabgkHiTiaadIhadaqhaaWcbaGaaGimaaqaaiaaiodaaa % GccqGHsislcaaIZaGaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjk % aiaawMcaamaaCaaaleqabaGaaGOmaaaakiabg2da9iaaigdacaaI3a % Gaeyi1HSTaamiEamaaDaaaleaacaaIWaaabaGaaGOnaaaakiabgUca % RiaaiAdacaWG4bWaa0baaSqaaiaaicdaaeaacaaI0aaaaOGaey4kaS % IaaGymaiaaicdacaWG4bWaa0baaSqaaiaaicdaaeaacaaIYaaaaOGa % eyOeI0IaaGymaiaaiEdacqGH9aqpcaaIWaaaaa!6633! AM = \sqrt {17} \Leftrightarrow A{M^2} = 17 \Leftrightarrow x_0^2 + {\left( { - x_0^3 - 3{x_0}} \right)^2} = 17 \Leftrightarrow x_0^6 + 6x_0^4 + 10x_0^2 - 17 = 0\)
\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaam % iEamaaDaaaleaacaaIWaaabaGaaGOmaaaakiabg2da9iaaigdacqGH % uhY2caWG4bWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaeyySaeRaaG % ymaiaac6caaaa!4565! \Leftrightarrow x_0^2 = 1 \Leftrightarrow {x_0} = \pm 1.\)
Vậy có hai điểm thỏa yêu cầu bài toán.