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Tentukan titik stasioner dan jenisnya untuk fungsi !

Pertanyaan

Tentukan titik stasioner dan jenisnya untuk fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator x squared over denominator x minus 1 end fraction end style!

Pembahasan Soal:

Titik stasioner atau titik balik suatu fungsi baik fungsi balik minimum maupun fungsi balik minimum.

Menentukan titik stasioner suatu fungsi begin mathsize 14px style y equals f left parenthesis x right parenthesis end style merupakan salah satu aplikasi turunan. Syarat stasioner adalah begin mathsize 14px style f apostrophe left parenthesis x right parenthesis equals 0 end style

Dari syarat stasioner terlebut diperoleh nilai undefined yang memenuhi misalkan begin mathsize 14px style x equals c end style maka titik stasionernya adalah begin mathsize 14px style open parentheses c comma f open parentheses c close parentheses close parentheses end style. Kemudian untuk menentukan jenis titik stasioner digunakan turunan tingkat 2 yaitu

  • Jika begin mathsize 14px style f " left parenthesis c right parenthesis greater than 0 end style merupakan titik balik minimum.
  • Jika begin mathsize 14px style f " left parenthesis c right parenthesis equals 0 end style merupakan titik belok
  • Jika begin mathsize 14px style f " left parenthesis c right parenthesis less than 0 end style merupakan titik balik maksimum.

Diketahui begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator x squared over denominator x minus 1 end fraction end style, misalkan begin mathsize 14px style y equals f left parenthesis x right parenthesis end style. Kemudian dicari terlebih dahulu begin mathsize 14px style f apostrophe open parentheses x close parentheses end style.

Diingat :begin mathsize 14px style open square brackets fraction numerator u left parenthesis x right parenthesis over denominator v left parenthesis x right parenthesis end fraction close square brackets to the power of apostrophe equals fraction numerator u apostrophe left parenthesis x right parenthesis times v left parenthesis x right parenthesis minus u left parenthesis x right parenthesis times v apostrophe left parenthesis x right parenthesis over denominator v left parenthesis x right parenthesis squared end fraction end style  maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell u left parenthesis x right parenthesis end cell equals cell x squared rightwards double arrow u apostrophe left parenthesis x right parenthesis equals 2 x end cell row cell v left parenthesis x right parenthesis end cell equals cell x minus 1 rightwards double arrow v apostrophe left parenthesis x right parenthesis equals 1 end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator open parentheses 2 x close parentheses open parentheses x minus 1 close parentheses minus open parentheses x squared close parentheses open parentheses 1 close parentheses over denominator open parentheses x minus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator 2 x squared minus 2 x minus x squared over denominator x squared minus x minus x plus 1 end fraction end cell row blank equals cell fraction numerator x squared minus 2 x over denominator x squared minus 2 x plus 1 end fraction end cell end table end style 

Diperoleh begin mathsize 14px style f apostrophe left parenthesis x right parenthesis equals fraction numerator x squared minus 2 x over denominator x squared minus 2 x plus 1 end fraction end style.

Kemudian dicari nilai undefined yang memenuhi undefined 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals 0 row blank left right double arrow cell fraction numerator x squared minus 2 x over denominator left right double arrow x squared minus 2 x plus 1 end fraction equals 0 comma x not equal to 1 end cell row blank left right double arrow cell x squared minus 2 x equals 0 end cell row blank left right double arrow cell x open parentheses x minus 2 close parentheses equals 0 end cell row blank left right double arrow cell x equals 0 logical or x equals 2 end cell end table end style 

Diperoleh begin mathsize 14px style x equals 0 space dan space x equals 2 end style dicari pasangan nilai x nya

untuk begin mathsize 14px style x equals 0 end style diperoleh 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f open parentheses 0 close parentheses end cell row blank equals cell fraction numerator 0 squared over denominator negative 1 end fraction end cell row blank equals 0 end table end style 

Diperoleh titik stasioner begin mathsize 14px style open parentheses 0 comma 0 close parentheses end style.

untuk begin mathsize 14px style x equals 2 end style diperoleh 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f left parenthesis 2 right parenthesis end cell row blank equals cell fraction numerator 2 squared over denominator 2 minus 1 end fraction end cell row blank equals 4 end table end style  

Diperoleh titik stasioner begin mathsize 14px style left parenthesis 2 comma 4 right parenthesis end style.

Kemudian dicari jenis titik stasioner yang ada, sebelumnya dicari turunan keduanya terlebih dahulu

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell u left parenthesis x right parenthesis end cell equals cell x squared minus 2 x rightwards double arrow u apostrophe left parenthesis x right parenthesis equals 2 x minus 2 end cell row cell v left parenthesis x right parenthesis end cell equals cell x squared minus 2 x plus 1 rightwards double arrow v apostrophe left parenthesis x right parenthesis equals 2 x minus 2 end cell row cell f " left parenthesis x right parenthesis end cell equals cell fraction numerator open parentheses 2 x minus 2 close parentheses open parentheses x squared minus 2 x plus 1 close parentheses minus open parentheses x squared minus 2 x close parentheses open parentheses 2 x minus 2 close parentheses over denominator open parentheses x squared minus 2 x plus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 2 x minus 2 close parentheses open parentheses x squared minus 2 x plus 1 minus open parentheses x squared minus 2 x close parentheses close parentheses over denominator open parentheses x minus 1 close parentheses to the power of 4 end fraction end cell row blank equals cell fraction numerator 2 up diagonal strike open parentheses x minus 1 close parentheses end strike open parentheses 1 close parentheses over denominator open parentheses x minus 1 close parentheses cubed up diagonal strike open parentheses x minus 1 close parentheses end strike end fraction end cell row blank equals cell 2 over open parentheses x minus 1 close parentheses cubed end cell end table end style 

Diperoleh begin mathsize 14px style f " left parenthesis x right parenthesis equals 2 over open parentheses x minus 1 close parentheses cubed end style.

untuk begin mathsize 14px style x equals 0 end style diperoleh 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f " left parenthesis 0 right parenthesis end cell equals cell 2 over open parentheses negative 1 close parentheses cubed end cell row blank equals cell negative 2 end cell row blank less than 0 end table end style 

Diperoleh titik stasioner begin mathsize 14px style open parentheses 0 comma 0 close parentheses end style adalah titik balik maksimum.

untuk begin mathsize 14px style x equals 2 end style diperoleh 

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f " left parenthesis 2 right parenthesis end cell equals cell 2 over open parentheses 2 minus 1 close parentheses cubed end cell row blank equals 2 row blank greater than 0 end table end style 

Diperoleh titik stasioner begin mathsize 14px style left parenthesis 2 comma 4 right parenthesis end style adalah titik balik minimum.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 02 April 2021

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