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Tentukan interval fungsi turun dari begin mathsize 14px style y equals 1 third x cubed minus 3 x squared plus 5 x plus 3 end style!

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S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

interval fungsi turun fungsi begin mathsize 14px style y equals 1 third x cubed minus 3 x squared plus 5 x plus 3 end style adalah begin mathsize 14px style 1 less than x less than 5 end style.

Pembahasan

Jika begin mathsize 14px style f to the power of apostrophe open parentheses x close parentheses end style bertanda negatif atau undefined maka kurva fungsi dalam keadaan turun (disebut fungsi turun).

Tentukan terlebih dahulu turunan dari fungsi begin mathsize 14px style y equals 1 third x cubed minus 3 x squared plus 5 x plus 3 end style seperti berikut:

begin mathsize 14px style table attributes columnalign center center left end attributes row cell y to the power of apostrophe end cell equals cell 1 third times 3 x to the power of 3 minus 1 end exponent minus 3 times 2 x to the power of 2 minus 1 end exponent plus 5 x to the power of 1 minus 1 end exponent plus 0 end cell row space equals cell 3 over 3 x squared minus 6 x to the power of 1 plus 5 x to the power of 0 end cell row space equals cell x squared minus 6 x to the power of 1 plus 5 open parentheses 1 close parentheses end cell row space equals cell x squared minus 6 x plus 5 end cell end table end style 

Kemudian diberi syarat begin mathsize 14px style y to the power of apostrophe less than 0 end style seperti berikut:

begin mathsize 14px style table attributes columnalign right center center end attributes row cell y to the power of apostrophe end cell less than 0 row cell x squared minus 6 x plus 5 end cell less than 0 row cell left parenthesis x minus 5 right parenthesis left parenthesis x minus 1 right parenthesis end cell less than 0 end table end style

Jika begin mathsize 14px style a times b less than 0 end style, maka dipenuhi salah satu:

(a) begin mathsize 14px style a less than 0 end style dan begin mathsize 14px style b greater than 0 end style 

(b) begin mathsize 14px style a greater than 0 end style dan begin mathsize 14px style b less than 0 end style 

(a) begin mathsize 14px style table attributes columnalign right center center end attributes row cell x minus 5 end cell less than 0 row x less than 5 end table end style dan begin mathsize 14px style table attributes columnalign right center center end attributes row cell x minus 1 end cell greater than 0 row x greater than 1 end table end style 

(b) begin mathsize 14px style table attributes columnalign right center center end attributes row cell x minus 5 end cell greater than 0 row x greater than 5 end table end style  dan begin mathsize 14px style table attributes columnalign right center center end attributes row cell x minus 1 end cell less than 0 row x less than 1 end table end style 

Kemudian didapat himpunan begin mathsize 14px style x end style, yaitu begin mathsize 14px style 1 less than x less than 5 end style

Jadi, interval fungsi turun fungsi begin mathsize 14px style y equals 1 third x cubed minus 3 x squared plus 5 x plus 3 end style adalah begin mathsize 14px style 1 less than x less than 5 end style.

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