Pertanyaan

Tentukan interval x yang menyebabkan fungsi f ( x ) = cos ( 2 x + 3 1 π ) dengan 0 ≤ x ≤ π a.fungsi naik b. fungsi turun

Tentukan interval $x$ yang menyebabkan fungsi $f (x) = cos (2 x + \frac{1}{3} π)$ dengan $0 \leq x \leq π$

a.fungsi naik
b. fungsi turun

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Jawaban

Fungsi naik pada Fungsi turun pada .

Fungsi

naik pada

0 less or equal than x less or equal than straight pi over 3

Fungsi

turun pada $straight pi over 3 less or equal than x less or equal than fraction numerator 5 straight pi over denominator 6 end fraction$ .

Pembahasan

Fungsi naik apabila Fungsi turun apabila Fungsi tetap atau statis apabila Tentukan turunan pertama fungsi Cari turunan Maka, Diketahui . Tentukan pembuat nol fungsi Uji nilai fungsi pada garis bilangan, a.fungsi naik Fungsi naik apabila , fungsi naik pada atau . b. fungsi turun Fungsi turun apabila , fungsi turun pada atau . Jadi,Fungsi naik pada Fungsi turun pada .

Fungsi naik apabila
Fungsi turun apabila
Fungsi tetap atau statis apabila

Tentukan turunan pertama fungsi f left parenthesis x right parenthesis equals cos open parentheses 2 x plus 1 third straight pi close parentheses

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos left parenthesis s plus t right parenthesis end cell equals cell cos left parenthesis s right parenthesis c o s left parenthesis t right parenthesis minus sin left parenthesis s right parenthesis sin left parenthesis t right parenthesis end cell row s equals cell 2 x end cell row t equals cell 1 third pi end cell row cell cos open parentheses 2 x plus 1 third pi close parentheses end cell equals cell cos open parentheses 2 x close parentheses cos open parentheses 1 third pi close parentheses minus sin open parentheses 2 x close parentheses sin open parentheses 1 third pi close parentheses end cell row blank equals cell cos open parentheses pi over 3 close parentheses cos open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction sin open parentheses 2 x close parentheses end cell row blank equals cell 1 half cos open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction sin open parentheses 2 x close parentheses end cell end table$

Cari turunan $fraction numerator d over denominator d x end fraction open parentheses 1 half cos open parentheses 2 x close parentheses close parentheses minus fraction numerator d over denominator d x end fraction open parentheses fraction numerator square root of 3 over denominator 2 end fraction sin open parentheses 2 x close parentheses close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses 1 half cos open parentheses 2 x close parentheses close parentheses end cell equals cell 1 half fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 2 x close parentheses close parentheses end cell row blank equals cell 1 half fraction numerator d over denominator d u end fraction open parentheses cos open parentheses u close parentheses close parentheses fraction numerator d over denominator d x end fraction open parentheses 2 x close parentheses end cell row blank equals cell 1 half open parentheses negative sin open parentheses 2 x close parentheses times thin space 2 close parentheses end cell row blank equals cell negative sin open parentheses 2 x close parentheses end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses fraction numerator square root of 3 over denominator 2 end fraction sin open parentheses 2 x close parentheses close parentheses end cell equals cell fraction numerator square root of 3 over denominator 2 end fraction fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 2 x close parentheses close parentheses end cell row blank equals cell fraction numerator square root of 3 over denominator 2 end fraction cos open parentheses 2 x close parentheses fraction numerator d over denominator d x end fraction open parentheses 2 x close parentheses end cell row blank equals cell fraction numerator square root of 3 over denominator 2 end fraction cos open parentheses 2 x close parentheses times thin space 2 end cell row blank equals cell square root of 3 cos open parentheses 2 x close parentheses end cell end table$

Maka,

$begin mathsize 12px style fraction numerator d over denominator d x end fraction open parentheses 1 half cos open parentheses 2 x close parentheses close parentheses minus fraction numerator d over denominator d x end fraction open parentheses fraction numerator square root of 3 over denominator 2 end fraction sin open parentheses 2 x close parentheses close parentheses equals negative sin open parentheses 2 x close parentheses minus square root of 3 cos open parentheses 2 x close parentheses end style$

Diketahui 0 less or equal than x less or equal than straight pi .

Tentukan pembuat nol fungsi f apostrophe left parenthesis x right parenthesis equals 0

$table attributes columnalign right center left columnspacing 0px end attributes row cell negative sin open parentheses 2 x close parentheses minus square root of 3 cos open parentheses 2 x close parentheses end cell equals 0 row cell negative sin left parenthesis 2 x right parenthesis end cell equals cell square root of 3 cos left parenthesis 2 x right parenthesis end cell row cell fraction numerator negative sin left parenthesis 2 x right parenthesis over denominator cos left parenthesis 2 x right parenthesis end fraction end cell equals cell square root of 3 end cell row cell negative tan left parenthesis 2 x right parenthesis end cell equals cell square root of 3 end cell row x equals 60 row x equals 150 end table$

Uji nilai fungsi f apostrophe left parenthesis x right parenthesis pada garis bilangan,

$table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell negative sin open parentheses 2 x close parentheses minus square root of 3 cos open parentheses 2 x close parentheses end cell row cell f left parenthesis 30 right parenthesis end cell equals cell negative sin left parenthesis 60 right parenthesis minus square root of 3 cos left parenthesis 60 right parenthesis end cell row blank equals cell negative fraction numerator square root of 3 over denominator 2 end fraction minus square root of 3 1 half end cell row blank equals cell negative square root of 3 end cell row cell f apostrophe left parenthesis x right parenthesis end cell less than 0 row cell f left parenthesis 120 right parenthesis end cell equals cell negative sin left parenthesis 240 right parenthesis minus square root of 3 cos left parenthesis 240 right parenthesis end cell row blank equals cell negative open parentheses negative fraction numerator square root of 3 over denominator 2 end fraction close parentheses minus square root of 3 open parentheses negative 1 half close parentheses end cell row blank equals cell square root of 3 end cell row cell f apostrophe left parenthesis x right parenthesis end cell greater than 0 end table$

a.fungsi naik

Fungsi f left parenthesis x right parenthesis naik apabila f apostrophe left parenthesis x right parenthesis greater than 0 , fungsi naik pada open square brackets 0 degree comma 60 degree close square brackets atau 0 less or equal than x less or equal than straight pi over 3 .

b. fungsi turun

Fungsi f left parenthesis x right parenthesis turun apabila , fungsi turun pada open square brackets 60 degree comma 150 degree close square brackets atau $straight pi over 3 less or equal than x less or equal than fraction numerator 5 straight pi over denominator 6 end fraction$ .

Jadi, Fungsi f left parenthesis x right parenthesis naik pada 0 less or equal than x less or equal than straight pi over 3 Fungsi turun pada $straight pi over 3 less or equal than x less or equal than fraction numerator 5 straight pi over denominator 6 end fraction$ .