Tentukan interval x sehingga grafik f(x)=2 sin x+cos 2x naik untuk 0∘≤x≤180∘!

Pertanyaan

Tentukan interval $x$ sehingga grafik $f(x)=2\;\sin\;x+\cos\;2x$ naik untuk $0^\circ\leq x\leq180^\circ$ !

N. Sari

Master Teacher

Mahasiswa/Alumni Universitas Nasional

Jawaban terverifikasi

Jawaban

grafik

f left parenthesis x right parenthesis equals 2 space sin space x plus cos space 2 x

naik pada interval 150 degree less than x less or equal than 180 degree

Pembahasan

Naik turunnya suatu tungsi dalam suatu interval tertentu dapat dilihat dari turunan pertamanya.

Jika , fungsi selalu naik.
Jika , fungsi selalu turun.
Jika , fungsi tidak naik dan tidak turun.

Diketahui f left parenthesis x right parenthesis equals 2 space sin space x plus cos space 2 x , maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 2 space sin space x plus cos space 2 x end cell row blank equals cell 2 space sin space x plus 1 minus 2 space sin squared space x end cell end table

Turunan pertamanya menjadi f apostrophe left parenthesis x right parenthesis equals 2 space cos space x minus 4 space sin space x times cos space x .

Fungsi naik ketika f apostrophe left parenthesis x right parenthesis greater than 0 , maka :

$table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell greater than 0 row cell 2 space cos space x minus 4 space sin space x times cos space x end cell greater than 0 row cell fraction numerator 2 space cos space x over denominator cos space x end fraction minus fraction numerator 4 space sin space x times cos space x over denominator cos space x end fraction end cell greater than cell fraction numerator 0 over denominator cos space x end fraction end cell row cell 2 minus 4 space sin space x end cell greater than 0 row cell negative 4 space sin space x end cell greater than cell negative 2 end cell row cell sin space x end cell less than cell fraction numerator negative 2 over denominator negative 4 end fraction end cell row cell sin space x end cell less than cell 1 half end cell end table$

Nilai perbandingan sinus bertanda negatif di kuadran II dan III sehingga :

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 150 degree end cell less than cell sin space x less than sin space 210 degree end cell row cell 150 degree end cell less than cell space space space x space space space less than 210 degree end cell end table

Karena fungsi memiliki batas 0 degree less or equal than x less or equal than 180 degree , maka interval positif terdapat pada 150 degree less than x less or equal than 180 degree .

Sehingga grafik f left parenthesis x right parenthesis equals 2 space sin space x plus cos space 2 x naik pada interval 150 degree less than x less or equal than 180 degree .

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