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Pertanyaan

Tentukan interval naik, interval turun, nilai stasioner dan titik stasioner:

begin mathsize 14px style g left parenthesis x right parenthesis equals 3 x squared plus 12 x minus 5 end style  

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Jawaban

untuk table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table, nilai stasionernya table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 17 end table dan titik stasionernya adalah begin mathsize 14px style left parenthesis negative 2 comma space minus 17 right parenthesis end style.

Pembahasan

Diketahui begin mathsize 14px style g left parenthesis x right parenthesis equals 3 x squared plus 12 x minus 5 end style, sehingga turunan pertama dari begin mathsize 14px style g left parenthesis x right parenthesis end style adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell 3 x squared plus 12 x minus 5 end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell 3 times 2 x plus 12 end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell 6 x plus 12 end cell end table end style  

Kurva table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis end cell end table akan selalu naik jika begin mathsize 14px style f apostrophe left parenthesis x right parenthesis greater than 0 end style.

Maka,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe left parenthesis x right parenthesis end cell greater than 0 row cell 6 x plus 12 end cell greater than 0 row cell 6 x end cell greater than cell negative 12 end cell row x greater than cell fraction numerator negative 12 over denominator 6 end fraction end cell row x greater than cell negative 2 end cell end table end style  

Dengan demikian, interval naiknya adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px greater than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table.

Kurva table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis end cell end table akan selalu turun jika begin mathsize 14px style f apostrophe left parenthesis x right parenthesis less than 0 end style.

Maka,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe left parenthesis x right parenthesis end cell less than 0 row cell 6 x plus 12 end cell less than 0 row cell 6 x end cell less than cell negative 12 end cell row x less than cell fraction numerator negative 12 over denominator 6 end fraction end cell row x less than cell negative 2 end cell end table end style   

Dengan demikian, interval turunnya adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table.

Titik stasioner merupakan titik dimana suatu fungsi berhenti naik atau turun.

Syarat stasioner adalah:

begin mathsize 14px style f apostrophe left parenthesis x right parenthesis equals 0 end style 

Sehingga,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe left parenthesis x right parenthesis end cell equals 0 row cell 6 x plus 12 end cell equals 0 row cell 6 x end cell equals cell negative 12 end cell row x equals cell fraction numerator negative 12 over denominator 6 end fraction end cell row x equals cell negative 2 end cell end table end style  

Untuk menentukan nilai stasioner, substitusikan table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table pada begin mathsize 14px style g left parenthesis x right parenthesis end style. Sehingga,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell 3 x squared plus 12 x minus 5 end cell row cell g left parenthesis negative 2 right parenthesis end cell equals cell 3 times open parentheses negative 2 close parentheses squared plus 12 times open parentheses negative 2 close parentheses minus 5 end cell row blank equals cell 3 times 4 minus 24 minus 5 end cell row blank equals cell 12 minus 24 minus 5 end cell row blank equals cell 12 minus 29 end cell row blank equals cell negative 17 end cell end table end style   

Dengan demikian, untuk table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 2 end table, nilai stasionernya table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 17 end table dan titik stasionernya adalah begin mathsize 14px style left parenthesis negative 2 comma space minus 17 right parenthesis end style.

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