Pertanyaan

Diketahui fungsi h ( x ) = 5 − 4 cos 2 3 x dengan − π < x < π . Tentukan beberapa hal berikut ini. 3. Interval ketika fungsi tersebut cekung ke bawah. ·

Diketahui fungsi $h (x) = 5 - 4 cos^{2} 3 x$ dengan $- π < x < π$ . Tentukan beberapa hal berikut
ini.

3. Interval ketika fungsi tersebut cekung ke bawah. ·

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Jawaban

fungsi dengan cekung ke bawah pada interval atau atau .

fungsi

space h left parenthesis x right parenthesis equals 5 minus 4 cos squared 3 x

dengan

negative straight pi less than straight x less than straight pi

cekung ke bawah pada interval straight pi over 12 less than straight x less than 1 fourth straight pi

atau

75 degree less than x less than 135 degree

atau

135 degree less than x less than 165 degree

Pembahasan

Turunan pertama, Turunan kedua, Syarat titik belok Hitung nilai Uji nilai fungsi pada garis bilangan, Fungsi cekung ke bawah apabila . Jadi,fungsi dengan cekung ke bawah pada interval atau atau .

Turunan pertama,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses 5 minus 4 cos squared open parentheses 3 x close parentheses close parentheses end cell equals cell fraction numerator d over denominator d x end fraction open parentheses 5 close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 4 cos squared open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 0 minus 4 fraction numerator d over denominator d x end fraction open parentheses cos squared open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 0 minus 4 fraction numerator d over denominator d u end fraction open parentheses u squared close parentheses fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 0 minus 4 open parentheses 2 u fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses close parentheses end cell row blank equals cell 0 minus 4 open parentheses 2 cos open parentheses 3 x close parentheses fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses close parentheses end cell row blank equals cell 0 minus 4 times thin space 2 cos open parentheses 3 x close parentheses open parentheses negative sin open parentheses 3 x close parentheses times thin space 3 close parentheses end cell row blank equals cell 0 minus open parentheses negative 24 cos open parentheses 3 x close parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 0 plus 24 cos open parentheses 3 x close parentheses sin open parentheses 3 x close parentheses end cell row blank equals cell 24 times fraction numerator sin open parentheses 2 times thin space 3 x close parentheses over denominator 2 end fraction end cell row blank equals cell 24 times fraction numerator sin open parentheses 6 x close parentheses over denominator 2 end fraction end cell row blank equals cell 12 sin open parentheses 6 x close parentheses end cell end table$

Turunan kedua,

$table attributes columnalign right center left columnspacing 0px end attributes row f equals cell sin left parenthesis u right parenthesis end cell row u equals cell 6 x end cell row cell fraction numerator d over denominator d x end fraction open parentheses 12 sin open parentheses 6 x close parentheses close parentheses end cell equals cell 12 fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 6 x close parentheses close parentheses end cell row blank equals cell 12 fraction numerator d over denominator d u end fraction open parentheses sin open parentheses u close parentheses close parentheses fraction numerator d over denominator d x end fraction open parentheses 6 x close parentheses end cell row blank equals cell 12 cos open parentheses u close parentheses fraction numerator d over denominator d x end fraction open parentheses 6 x close parentheses end cell row blank equals cell 12 cos open parentheses 6 x close parentheses fraction numerator d over denominator d x end fraction open parentheses 6 x close parentheses end cell row blank equals cell 12 cos open parentheses 6 x close parentheses times thin space 6 end cell row blank equals cell 72 cos open parentheses 6 x close parentheses end cell end table$

Syarat titik belok h apostrophe apostrophe left parenthesis x right parenthesis equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell 72 cos 6 x end cell equals 0 row x equals 15 row x equals 75 row x equals 135 row x equals 45 row x equals 105 row x equals 165 end table

Hitung nilai h left parenthesis x right parenthesis

$table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell 5 minus 4 cos squared 3 x end cell row cell h left parenthesis 15 right parenthesis end cell equals cell 5 minus 4 cos squared 45 end cell row blank equals cell 5 minus 4 open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared end cell row blank equals 3 row cell h left parenthesis 75 right parenthesis end cell equals cell 5 minus 4 cos squared 225 end cell row blank equals cell 5 minus 4 times fraction numerator 1 plus cos open parentheses 2 times thin space 225 to the power of ring operator thin space end exponent close parentheses over denominator 2 end fraction end cell row blank equals cell cos open parentheses 1 half 180 to the power of ring operator thin space end exponent close parentheses end cell row blank equals cell negative 2 times thin space 0 plus 3 end cell row blank equals 3 row blank blank blank row cell h left parenthesis 135 right parenthesis end cell equals cell 5 minus 4 cos squared 405 end cell row blank equals cell 5 minus 4 cos squared open parentheses 45 to the power of ring operator thin space end exponent close parentheses end cell row blank equals cell 5 minus 4 open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared end cell row blank equals 3 row blank blank blank row cell h left parenthesis 45 right parenthesis end cell equals cell 5 minus 4 cos squared 135 end cell row blank equals cell 5 minus 4 open parentheses negative sin open parentheses 45 to the power of ring operator thin space end exponent close parentheses close parentheses squared end cell row blank equals cell 5 minus 4 open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared end cell row blank equals 3 row blank blank blank row cell h left parenthesis 105 right parenthesis end cell equals cell 5 minus 4 cos squared 315 end cell row blank equals cell 5 minus 4 sin squared open parentheses 225 to the power of ring operator thin space end exponent close parentheses end cell row blank equals cell 5 minus 4 open parentheses negative fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared end cell row blank equals 3 row blank blank blank row cell h left parenthesis 165 right parenthesis end cell equals cell 5 minus 4 cos squared 495 end cell row blank equals cell 5 minus 4 sin squared open parentheses 90 to the power of ring operator thin space end exponent minus 495 to the power of ring operator thin space end exponent close parentheses end cell row blank equals cell 5 minus 4 open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared end cell row blank equals 3 end table$

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

Fungsi f left parenthesis x right parenthesis cekung ke bawah apabila f apostrophe apostrophe left parenthesis x right parenthesis less than 0 .

Jadi, fungsi space h left parenthesis x right parenthesis equals 5 minus 4 cos squared 3 x dengan negative straight pi less than straight x less than straight pi cekung ke bawah pada interval straight pi over 12 less than straight x less than 1 fourth straight pi atau 75 degree less than x less than 135 degree atau 135 degree less than x less than 165 degree .