Pertanyaan

Tentukan titik balik dari f ( x ) = x 4 – x 2

Tentukan titik balik dari $f (x) = x^{4} - x^{2}$

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F. Ayudhita

Master Teacher

Jawaban terverifikasi

Jawaban

titik balik untuk ada tiga, yaitu , , dan .

titik balik untuk

ada tiga, yaitu

, $begin mathsize 14px style open parentheses fraction numerator square root of 2 over denominator 2 end fraction comma negative 1 fourth close parentheses end style$ , dan $begin mathsize 14px style open parentheses negative fraction numerator square root of 2 over denominator 2 end fraction comma negative 1 fourth close parentheses end style$ .

Pembahasan

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

Titik balik suatu fungsi di , berlaku jika dan

Dicari terlebih dahulu turunan dari

begin mathsize 14px style f open parentheses x close parentheses equals x to the power of 4 minus x squared f to the power of apostrophe open parentheses x close parentheses equals 4 x cubed minus 2 x f to the power of double apostrophe open parentheses x close parentheses equals 12 x squared minus 2 end style

Selanjutnya

$begin mathsize 14px style f to the power of apostrophe open parentheses x close parentheses equals 4 x cubed minus 2 x equals 0 2 x open parentheses 2 x squared minus 1 close parentheses equals 0 x equals 0 comma space x equals plus-or-minus fraction numerator square root of 2 over denominator 2 end fraction end style$

Kemudian saat $begin mathsize 14px style x equals 0 comma space x equals plus-or-minus fraction numerator square root of 2 over denominator 2 end fraction end style$ , sehingga syarat terpenuhi.

Substitusikan ke

Untuk

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 to the power of 4 minus x squared end cell row cell f open parentheses 0 close parentheses end cell equals cell 0 to the power of 4 minus 0 squared end cell row blank equals 0 end table end style

Untuk $begin mathsize 14px style x equals fraction numerator square root of 2 over denominator 2 end fraction end style$

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x to the power of 4 minus x squared end cell row cell f open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses end cell equals cell open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses to the power of 4 minus open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared end cell row blank equals cell 4 over 16 minus 2 over 4 end cell row blank equals cell fraction numerator 4 minus 8 over denominator 16 end fraction end cell row blank equals cell negative 1 fourth end cell end table end style$

Untuk $begin mathsize 14px style x equals negative fraction numerator square root of 2 over denominator 2 end fraction end style$

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x to the power of 4 minus x squared end cell row cell f open parentheses negative fraction numerator square root of 2 over denominator 2 end fraction close parentheses end cell equals cell open parentheses negative fraction numerator square root of 2 over denominator 2 end fraction close parentheses to the power of 4 minus open parentheses negative fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared end cell row blank equals cell 4 over 16 minus 2 over 4 end cell row blank equals cell fraction numerator 4 minus 8 over denominator 16 end fraction end cell row blank equals cell negative 1 fourth end cell end table end style$

Jadi titik balik untuk ada tiga, yaitu , $begin mathsize 14px style open parentheses fraction numerator square root of 2 over denominator 2 end fraction comma negative 1 fourth close parentheses end style$ , dan $begin mathsize 14px style open parentheses negative fraction numerator square root of 2 over denominator 2 end fraction comma negative 1 fourth close parentheses end style$ .