Pertanyaan

Tentukan titik balik dari f ( x ) = x 3 − x

Tentukan titik balik dari $f (x) = x^{3} - x$

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F. Ayudhita

Master Teacher

Jawaban terverifikasi

Jawaban

titik balik untuk ada dua, yaitu dan

titik balik untuk

ada dua, yaitu $begin mathsize 14px style open parentheses fraction numerator square root of 3 over denominator 3 end fraction comma negative fraction numerator 2 square root of 3 over denominator 9 end fraction close parentheses end style$ dan $begin mathsize 14px style open parentheses negative fraction numerator square root of 3 over denominator 3 end fraction comma fraction numerator 2 square root of 3 over denominator 9 end fraction close parentheses end style$

Pembahasan

Titik balik suatu fungsi di , berlaku jika dan Dicari terlebih dahulu turunan dari Selanjutnya Kemudian saat , sehingga syarat terpenuhi. Substitusikan ke Untuk Untuk Jadi titik balik untuk ada dua, yaitu dan

Titik balik suatu fungsi di , berlaku jika dan

Dicari terlebih dahulu turunan dari

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Selanjutnya

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of apostrophe open parentheses x close parentheses end cell equals cell 3 x squared minus 1 equals 0 end cell row cell 3 x squared end cell equals 1 row cell x squared end cell equals cell 1 third end cell row x equals cell plus-or-minus square root of 1 third end root end cell row blank equals cell plus-or-minus open parentheses fraction numerator 1 over denominator square root of 3 end fraction times fraction numerator square root of 3 over denominator square root of 3 end fraction close parentheses end cell row blank equals cell plus-or-minus fraction numerator square root of 3 over denominator 3 end fraction end cell end table end style$

Kemudian saat $begin mathsize 14px style x equals plus-or-minus fraction numerator square root of 3 over denominator 3 end fraction end style$ , sehingga syarat terpenuhi.

Substitusikan ke

Untuk $begin mathsize 14px style x equals fraction numerator square root of 3 over denominator 3 end fraction end style$

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x cubed minus x end cell row cell f open parentheses fraction numerator square root of 3 over denominator 3 end fraction close parentheses end cell equals cell open parentheses fraction numerator square root of 3 over denominator 3 end fraction close parentheses cubed minus fraction numerator square root of 3 over denominator 3 end fraction end cell row blank equals cell fraction numerator 3 square root of 3 over denominator 27 end fraction minus fraction numerator square root of 3 over denominator 3 end fraction end cell row blank equals cell fraction numerator 3 square root of 3 minus 9 square root of 3 over denominator 27 end fraction end cell row blank equals cell negative fraction numerator 6 square root of 3 over denominator 27 end fraction end cell row blank equals cell negative fraction numerator 2 square root of 3 over denominator 9 end fraction end cell end table end style$

Untuk $begin mathsize 14px style x equals negative fraction numerator square root of 3 over denominator 3 end fraction end style$

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x cubed minus x end cell row cell f open parentheses negative fraction numerator square root of 3 over denominator 3 end fraction close parentheses end cell equals cell open parentheses negative fraction numerator square root of 3 over denominator 3 end fraction close parentheses cubed minus open parentheses negative fraction numerator square root of 3 over denominator 3 end fraction close parentheses end cell row blank equals cell negative fraction numerator 3 square root of 3 over denominator 27 end fraction plus fraction numerator square root of 3 over denominator 3 end fraction end cell row blank equals cell fraction numerator negative 3 square root of 3 plus 9 square root of 3 over denominator 27 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 over denominator 27 end fraction end cell row blank equals cell fraction numerator 2 square root of 3 over denominator 9 end fraction end cell end table end style$

Jadi titik balik untuk ada dua, yaitu $begin mathsize 14px style open parentheses fraction numerator square root of 3 over denominator 3 end fraction comma negative fraction numerator 2 square root of 3 over denominator 9 end fraction close parentheses end style$ dan $begin mathsize 14px style open parentheses negative fraction numerator square root of 3 over denominator 3 end fraction comma fraction numerator 2 square root of 3 over denominator 9 end fraction close parentheses end style$