Pertanyaan

Selidiki titik belok kurva fungsi h ( x ) = 2 1 x − cos x ; 0 < x < 2 π .

Selidiki titik belok kurva fungsi $h (x) = \frac{1}{2} x - cos x$ ; $0 < x < 2 π$ .

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Jawaban terverifikasi

Jawaban

titik belok kurva fungsi adalah dan .

titik belok kurva fungsi h left parenthesis x right parenthesis equals 1 half x minus cos x

adalah

dan

Pembahasan

Untuk mencari titik belok, cari turunan fungsi. Turunan pertama, Turunan kedua, Syarat titik belok Hitung nilai Uji nilai fungsi pada garis bilangan, Titik atau dan atau merupakan titik belok, karena di titik dan terjadi perubahan kecekungan. Jadi, titik belok kurva fungsi adalah dan .

Untuk mencari titik belok, cari turunan fungsi.

Turunan pertama,

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

Turunan kedua,

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

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

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x end cell equals 0 row x equals 90 row x equals 270 end table

Hitung nilai f left parenthesis x right parenthesis

$table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 1 half x minus cos x end cell row cell f left parenthesis 90 right parenthesis end cell equals cell 1 half open parentheses straight pi over 2 close parentheses minus cos 90 end cell row blank equals cell straight pi over 4 minus 0 end cell row blank equals cell straight pi over 4 end cell row cell f left parenthesis 270 right parenthesis end cell equals cell 1 half open parentheses fraction numerator 3 straight pi over denominator 2 end fraction close parentheses minus cos 270 end cell row blank equals cell fraction numerator 3 straight pi over denominator 4 end fraction minus 0 end cell row blank equals cell fraction numerator 3 straight pi over denominator 4 end fraction end cell end table$

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

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

Titik open parentheses straight pi over 2 comma straight pi over 4 close parentheses atau open parentheses 90 degree comma straight pi over 4 close parentheses dan $open parentheses 270 degree comma fraction numerator 3 straight pi over denominator 4 end fraction close parentheses$ atau $open parentheses fraction numerator 3 straight pi over denominator 2 end fraction comma fraction numerator 3 straight pi over denominator 4 end fraction close parentheses$ merupakan titik belok, karena di titik x equals straight pi over 2 dan $x equals fraction numerator 3 straight pi over denominator 2 end fraction$ terjadi perubahan kecekungan.

Jadi, titik belok kurva fungsi h left parenthesis x right parenthesis equals 1 half x minus cos x adalah 1 half straight pi dan 3 over 2 straight pi .