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Jika periode fungsi  adalah , maka nilai maksimum ...

Jika periode fungsi f left parenthesis x right parenthesis equals a space sin space left parenthesis a x right parenthesis plus 4 adalah straight pi over 2, maka nilai maksimum fungsi f adalah...

  1. 4

  2. 6

  3. 8

  4. 12

  5. 16

Jawaban:

i) Nilai maksimum dan minimum pada fungsi trigonometri

f left parenthesis x right parenthesis space equals space sin space x space rightwards arrow space minus 1 space less or equal than space f left parenthesis x right parenthesis space less or equal than space 1  f left parenthesis x right parenthesis space equals space sin space a x space rightwards arrow space minus 1 space less or equal than space f left parenthesis x right parenthesis space less or equal than space 1  f left parenthesis x right parenthesis space equals space a sin space a x space rightwards arrow space minus vertical line a vertical line space less or equal than space f left parenthesis x right parenthesis space less or equal than space vertical line a vertical line  f left parenthesis x right parenthesis space equals space a sin space a x space plus space 4 space rightwards arrow space minus vertical line a vertical line space plus space 4 space less or equal than space f left parenthesis x right parenthesis space less or equal than space vertical line a vertical line space plus space 4

ii) Periode pada fungsi trigonometri

Fungsi space f left parenthesis x right parenthesis space equals space a space sin space b x space plus space c space memiliki space periode space fraction numerator 2 straight pi over denominator vertical line b vertical line end fraction.  Sehingga space fungsi space f left parenthesis x right parenthesis space equals space a space sin left parenthesis a x right parenthesis space plus space 4 space memiliki space periode space fraction numerator 2 straight pi over denominator vertical line a vertical line end fraction.

iii) Bandingkan periode fungsi

Dari perhitungan didapat periodefraction numerator 2 straight pi over denominator vertical line a vertical line end fraction.

Dari soal didapat periode straight pi over 2.

Maka,

fraction numerator 2 straight pi over denominator vertical line a vertical line end fraction space equals space straight pi over 2  vertical line a vertical line space equals space 4

iv) Hitung nilai minimum fungsi

Dari sebelumnya didapat nilai maksimum fungsi adalah |a| + 4, sehingga

maks f left parenthesis x right parenthesis space equals space vertical line a vertical line plus 4 space equals space 4 plus 4 equals space 8

 

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