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Diketahui fungsi  untuk 0≤x≤2π. Fungsi g naik pada interval ...

Pertanyaan

Diketahui fungsi g open parentheses x close parentheses equals cos invisible function application open parentheses x minus pi over 3 close parentheses untuk 0 less or equal than x less or equal than 2 pi. Fungsi g naik pada interval ...

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

dapat disimpulkan bahwa fungsi g naik pada interval 0 less than x less than pi over 3 dan fraction numerator 4 pi over denominator 3 end fraction less than x less than 2 pi.

Pembahasan

Diketahui: Fungsi g open parentheses x close parentheses equals cos invisible function application open parentheses x minus pi over 3 close parentheses untuk 0 less or equal than x less or equal than 2 pi

Ditanya: Fungsi g naik pada interval?

Jawab: 

Ingat bahwa fungsi g open parentheses x close parentheses dikatakan naik jika g apostrophe open parentheses x close parentheses greater than 0.

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x close parentheses end cell equals cell cos invisible function application open parentheses x minus pi over 3 close parentheses end cell row cell g apostrophe open parentheses x close parentheses end cell equals cell negative sin invisible function application open parentheses x minus pi over 3 close parentheses end cell end table

Didapatkan bahwa jika fungsi g naik apabila negative sin invisible function application open parentheses x minus pi over 3 close parentheses greater than 0 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell negative sin invisible function application open parentheses x minus pi over 3 close parentheses end cell greater than 0 row cell sin invisible function application open parentheses x minus pi over 3 close parentheses end cell less than 0 end table

Cari nilai x saat sin invisible function application open parentheses x minus pi over 3 close parentheses equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell sin invisible function application open parentheses x minus pi over 3 close parentheses end cell equals 0 row cell sin invisible function application open parentheses x minus pi over 3 close parentheses end cell equals cell sin invisible function application 0 end cell row cell x minus pi over 3 end cell equals 0 row x equals cell pi over 3 end cell end table

atau

table attributes columnalign right center left columnspacing 0px end attributes row cell sin invisible function application open parentheses x minus pi over 3 close parentheses end cell equals 0 row cell sin invisible function application open parentheses x minus pi over 3 close parentheses end cell equals cell sin invisible function application pi end cell row cell x minus pi over 3 end cell equals pi row x equals cell pi plus pi over 3 end cell row blank equals cell fraction numerator 3 pi over denominator 3 end fraction plus pi over 3 end cell row blank equals cell fraction numerator 4 pi over denominator 3 end fraction end cell end table

Didapatkan nilai x equals pi over 3 dan x equals fraction numerator 4 pi over denominator 3 end fraction. Kemudian uji nilai sin invisible function application open parentheses x minus pi over 3 close parentheses pada x equals pi over 6 sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell sin invisible function application open parentheses x minus pi over 3 close parentheses end cell equals cell sin invisible function application open parentheses pi over 6 minus pi over 3 close parentheses end cell row blank equals cell sin invisible function application open parentheses pi over 6 minus fraction numerator 2 pi over denominator 6 end fraction close parentheses end cell row blank equals cell sin invisible function application open parentheses negative pi over 6 close parentheses end cell row blank equals cell sin invisible function application open parentheses negative fraction numerator 180 degree over denominator 6 end fraction close parentheses end cell row blank equals cell sin invisible function application open parentheses negative 30 degree close parentheses end cell row blank equals cell negative sin invisible function application open parentheses 360 degree minus 30 degree close parentheses end cell row blank equals cell negative sin invisible function application open parentheses 30 degree close parentheses end cell row blank equals cell negative 1 half end cell end table

Diperoleh nilai sin invisible function application open parentheses x minus pi over 3 close parentheses pada x equals pi over 6 yaitu negative 1 half sehingga 

Karena syarat fungsi g naik apabila negative sin invisible function application open parentheses x minus pi over 3 close parentheses greater than 0 atau sin invisible function application open parentheses x minus pi over 3 close parentheses less than 0 maka intervalnya yaitu 0 less than x less than pi over 3 dan fraction numerator 4 pi over denominator 3 end fraction less than x less than 2 pi.

Jadi, dapat disimpulkan bahwa fungsi g naik pada interval 0 less than x less than pi over 3 dan fraction numerator 4 pi over denominator 3 end fraction less than x less than 2 pi.

441

2.6 (3 rating)

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