Tentukan titik balik dari f(x) = x3−x

Pertanyaan

Tentukan titik balik dari f italic left parenthesis x italic right parenthesis italic space italic equals italic space x to the power of italic 3 italic minus x

  1. (....)

  2. (....)

F. Ayudhita

Master Teacher

Jawaban terverifikasi

Jawaban

titik balik untuk undefined 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

Pembahasan

Titik balik suatu fungsi begin mathsize 14px style f end style di begin mathsize 14px style x equals c end style, berlaku jika begin mathsize 14px style f to the power of apostrophe left parenthesis c right parenthesis equals 0 end style dan begin mathsize 14px style f to the power of double apostrophe left parenthesis c right parenthesis not equal to 0 end style

Dicari terlebih dahulu turunan dari begin mathsize 14px style f open parentheses x close parentheses equals x cubed minus x end style

undefined

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 stylebegin mathsize 14px style f to the power of double apostrophe open parentheses x close parentheses equals 6 x not equal to 0 end style sehingga syarat terpenuhi.

Substitusikan undefined ke begin mathsize 14px style y equals f open parentheses x close parentheses equals x cubed minus x end style

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 undefined 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

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