Roboguru

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Diketahui f(x) = a cos (2x) + b dan g(x) = b sin (bx) + a. Jika f dan g mempunyai periode yang sama dan nilai minimum fungsi g adalah 0, nilai maksimum fungsi f adalah ...

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D. Kamilia

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Ingat factorial space Periode space pada space fungsi space straight y equals straight a space cos space bx plus straight c space adalah space fraction numerator 2 straight pi over denominator open vertical bar straight b close vertical bar end fraction.  Fungsi space straight f left parenthesis straight x right parenthesis equals straight a space cos invisible function application left parenthesis 2 straight x right parenthesis plus straight b space memiliki space periode space fraction numerator 2 straight pi over denominator open vertical bar 2 close vertical bar end fraction equals box enclose straight pi.  Sementara space fungsi space straight g left parenthesis straight x right parenthesis equals straight b space sin left parenthesis bx right parenthesis plus straight a space memiliki space periode space box enclose fraction numerator 2 straight pi over denominator open vertical bar straight b close vertical bar end fraction end enclose.  Karena space straight f space dan space straight g space memiliki space periode space yang space sama comma space maka  straight pi equals fraction numerator 2 straight pi over denominator open vertical bar straight b close vertical bar end fraction  open vertical bar straight b close vertical bar equals 2  box enclose straight b equals plus-or-minus 2 end enclose    straight g left parenthesis straight x right parenthesis equals straight b space sin invisible function application bx plus straight a rightwards arrow box enclose negative vertical line straight b vertical line plus straight a less or equal than straight g left parenthesis straight x right parenthesis less or equal than vertical line straight b vertical line plus straight a end enclose    Karena space minus vertical line straight b vertical line plus straight a less or equal than straight g left parenthesis straight x right parenthesis less or equal than vertical line straight b vertical line plus straight a comma space maka space nilai space minimum space untuk space straight g left parenthesis straight x right parenthesis space adalah space box enclose negative vertical line straight b vertical line plus straight a end enclose. space Dari space soal space diketahui space bahwa space nilai space minimum space untuk space straight g left parenthesis straight x right parenthesis space adalah space 0 comma space maka space  minus vertical line straight b vertical line plus straight a equals 0 space  straight a equals vertical line straight b vertical line space  box enclose straight a equals 2 end enclose    Untuk space straight b equals negative 2 space  straight f left parenthesis straight x right parenthesis equals straight a space cos invisible function application left parenthesis 2 straight x right parenthesis plus straight b space  straight f left parenthesis straight x right parenthesis equals 2 space cos invisible function application left parenthesis 2 straight x right parenthesis minus 2 space  Karena space minus 1 less or equal than cos invisible function application 2 straight x less or equal than 1 comma space maka space  minus 2 minus 2 less or equal than straight f left parenthesis straight x right parenthesis less or equal than 2 minus 2  minus 4 less or equal than straight f left parenthesis straight x right parenthesis less or equal than 0 space  Didapatkan space nilai space maksimum space untuk space straight f left parenthesis straight x right parenthesis space adalah space box enclose 0. space space    Untuk space straight b equals 2 space  straight f left parenthesis straight x right parenthesis equals straight a space cos invisible function application left parenthesis 2 straight x right parenthesis plus straight b  space straight f left parenthesis straight x right parenthesis equals 2 space cos invisible function application left parenthesis 2 straight x right parenthesis plus 2 space  Karena space minus 1 less or equal than cos invisible function application 2 straight x less or equal than 1 comma space maka space  minus 2 plus 2 less or equal than straight f left parenthesis straight x right parenthesis less or equal than 2 plus 2 space  0 less or equal than straight f left parenthesis straight x right parenthesis less or equal than 4 space  Didapatkan space nilai space maksimum space untuk space straight f left parenthesis straight x right parenthesis space adalah space box enclose 4. space space    Sehingga space secara space keseluruhan comma space nilai space maksimum space untuk space straight f left parenthesis straight x right parenthesis space adalah space box enclose 4.

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