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Fungsi  turun pada interval ...

Pertanyaan

Fungsi f open parentheses x close parentheses equals square root of sin squared space x plus open parentheses x over 2 close parentheses end root comma space x greater than 0 turun pada interval ...

Pembahasan Soal:

Fungsi turun ketika f apostrophe left parenthesis x right parenthesis less than 0.

Tentukan terlebih dahulu turunan fungsinya.

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell square root of sin squared space x plus open parentheses x over 2 close parentheses end root end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 1 half open parentheses sin squared space x plus open parentheses x over 2 close parentheses close parentheses to the power of 1 half minus 1 end exponent times open parentheses 2 space sin space x times cos space x plus 1 half close parentheses end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 1 half open parentheses sin squared space x plus open parentheses x over 2 close parentheses close parentheses to the power of negative 1 half end exponent times open parentheses sin space 2 x plus 1 half close parentheses end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator sin space 2 x plus 1 half over denominator square root of sin squared space x plus open parentheses x over 2 close parentheses end root end fraction end cell end table

Kemudian sustitusi syarat fungsi turun, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell less than 0 row cell fraction numerator sin space 2 x plus begin display style 1 half end style over denominator square root of sin squared space x plus open parentheses begin display style x over 2 end style close parentheses end root end fraction end cell less than 0 row cell sin space 2 x plus 1 half end cell less than 0 row cell sin space 2 x end cell less than cell negative 1 half end cell row blank blank blank end table

Perhatikan penyebutnya! Penyebut pecahan tersebut berbentuk akar. Bentuk akar selalu menghasilkan nilai positif atau definit positif sehingga dapat diabaikan.

Pembuat nol dari pertidaksamaan tersebut adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 210 degree plus straight k times 360 degree end cell row x equals cell 105 degree plus straight k times 180 degree end cell row blank equals cell... comma space minus 75 degree comma space 105 degree comma space 285 degree comma space... end cell row blank equals cell... comma space minus 5 over 12 straight pi comma space 7 over 12 straight pi comma space 19 over 12 straight pi comma space... end cell row blank blank blank row cell 2 x end cell equals cell 330 degree plus straight k times 360 degree end cell row x equals cell 165 degree plus straight k times 180 degree end cell row blank equals cell... comma space minus 15 degree comma space 165 degree comma space 345 degree comma space... end cell row blank equals cell... comma space minus 1 over 12 straight pi comma space 11 over 12 straight pi comma space 23 over 12 straight pi comma space... end cell end table

Maka, fungsi f open parentheses x close parentheses equals square root of sin squared space x plus open parentheses x over 2 close parentheses end root comma space x greater than 0 turun pada interval negative 15 over 12 straight pi less than straight x less than 23 over 12 straight pi.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 05 Mei 2021

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