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Tentukan titik maksimum dan minimum dari fungsi  pada selang .

Pertanyaan

Tentukan titik maksimum dan minimum dari fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals x plus cos space 2 x end style pada selang begin mathsize 14px style 0 less or equal than x less or equal than 180 degree end style.

Pembahasan Video:

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals 0 row cell 1 minus 2 times sin space 2 x end cell equals 0 row cell negative 2 times sin space 2 x end cell equals cell negative 1 end cell row cell sin space 2 x end cell equals cell 1 half end cell row cell sin space 2 x end cell equals cell sin space 30 degree end cell end table 

Note : 

Jika diketahui persamaan sin space x equals sin space alpha. maka penyelesaiannya adalah 

x equals alpha plus k times 360 degree atau x equals left parenthesis 180 degree minus alpha right parenthesis plus k times 360 degree

 Penyelesaian 1

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 30 degree plus k times 360 degree end cell row x equals cell 15 degree plus k times 180 degree end cell end table

k equals 0 rightwards arrow x equals 15 degree k equals 1 rightwards arrow x equals 195 degree space

karena saat k equals 1 menghasilkan nilai x diluar interval 0 less or equal than x less or equal than 180 degree maka tidak termasuk himpunan penyelesaian.

Penyelesaian 2

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell left parenthesis 180 degree minus 30 degree right parenthesis plus k times 360 degree end cell row cell 2 x end cell equals cell 150 degree plus k times 360 degree end cell row x equals cell 75 degree plus k times 360 degree end cell end table

k equals 0 rightwards arrow x equals 75 degree k equals 1 rightwards arrow x equals 255 degree

karena saat k equals 1 menghasilkan nilai x diluar interval 0 less or equal than x less or equal than 180 degree maka tidak termasuk himpunan penyelesaian.

Cek titik :

  • Untuk x equals 15 degree

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis 15 degree right parenthesis end cell equals cell 15 degree plus cos space 30 degree end cell row blank equals cell 15 degree plus fraction numerator square root of 3 over denominator 2 end fraction end cell row blank equals cell 1 over 12 straight pi plus fraction numerator square root of 3 over denominator 2 end fraction end cell row blank almost equal to cell 1.127 end cell end table

  • Untuk x equals 75 degree

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis 75 degree right parenthesis end cell equals cell 75 degree plus cos space 150 degree end cell row blank equals cell 75 degree minus fraction numerator square root of 3 over denominator 2 end fraction end cell row blank equals cell 5 over 12 straight pi minus fraction numerator square root of 3 over denominator 2 end fraction end cell row blank almost equal to cell 0.443 end cell end table

  • untuk x equals 0 degree

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis 0 degree right parenthesis end cell equals cell 0 degree plus cos space 0 degree end cell row blank equals cell 0 degree plus 1 end cell row blank equals 1 end table

  • untuk x equals 180 degree

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis 180 degree right parenthesis end cell equals cell 180 degree plus cos space 360 degree end cell row blank equals cell 180 degree plus 1 end cell row blank equals cell straight pi plus 1 end cell row blank almost equal to cell 4.1416 end cell end table

Titik maksimumnya adalah left parenthesis 180 degree comma space 4.1416 right parenthesis

Titik minimumnya adalah open parentheses 75 degree comma 0.443 close parentheses

Jadi, titik maksimum dan minimum dari fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals x plus cos space 2 x end style pada selang begin mathsize 14px style 0 less or equal than x less or equal than 180 degree end style berturut-turut adalah left parenthesis 180 degree comma space 4.1416 right parenthesis dan open parentheses 75 degree comma 0.443 close parentheses.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 04 Mei 2021

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