Roboguru

Diberikan . Misalkan  dan  adalah bilangan real positif yang memenuhi . Nilai minimum dari  adalah .... (Soal OSN-P Matematika SMA Tahun 2008)

Pertanyaan

Diberikan begin mathsize 14px style f left parenthesis x right parenthesis equals x squared plus 4 end style. Misalkan begin mathsize 14px style x end style dan begin mathsize 14px style y end style adalah bilangan real positif yang memenuhi begin mathsize 14px style f left parenthesis x y right parenthesis plus f left parenthesis y minus x right parenthesis equals f left parenthesis y plus x right parenthesis end style. Nilai minimum dari begin mathsize 14px style x plus y end style adalah ....

(Soal OSN-P Matematika SMA Tahun 2008)

Pembahasan Soal:

begin mathsize 14px style f left parenthesis x right parenthesis equals x squared plus 4 f left parenthesis x y right parenthesis equals x squared y squared plus 4 f left parenthesis y minus x right parenthesis equals open parentheses y minus x close parentheses squared plus 4 f left parenthesis y plus x right parenthesis equals open parentheses y plus x close parentheses squared plus 4 end style 
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x y right parenthesis plus f left parenthesis y minus x right parenthesis end cell equals cell f left parenthesis y plus x right parenthesis end cell row cell x squared y squared plus 4 plus open parentheses y minus x close parentheses squared plus up diagonal strike 4 end cell equals cell open parentheses y plus x close parentheses squared plus up diagonal strike 4 end cell row cell x squared y squared plus up diagonal strike y squared end strike plus up diagonal strike x squared end strike minus 2 x y plus 4 end cell equals cell up diagonal strike y squared end strike plus up diagonal strike x squared end strike plus 2 x y end cell row cell x squared y squared minus 2 x y plus 4 end cell equals cell 2 x y end cell row cell x squared y squared minus 4 x y plus 4 end cell equals 0 row cell open parentheses x y minus 2 close parentheses squared end cell equals 0 row cell x y end cell equals 2 row blank blank blank row blank blank blank end table end style 

Dengan ketaksamaan AM-GM.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator x plus y over denominator 2 end fraction end cell greater or equal than cell square root of x y end root end cell row cell fraction numerator x plus y over denominator 2 end fraction end cell greater or equal than cell square root of 2 end cell row cell x plus y end cell greater or equal than cell 2 square root of 2 end cell row blank blank blank end table end style 

Kuadrat tidak mungkin negatif.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x plus y end cell equals cell x plus y end cell row blank equals cell x plus 2 over x end cell row blank equals cell open parentheses square root of x minus fraction numerator square root of 2 over denominator square root of x end fraction close parentheses squared plus 2 square root of 2 end cell end table end style 

Sehingga begin mathsize 14px style x equals y equals square root of 2 end style

Jadi, nilai minimum dari begin mathsize 14px style x plus y end style adalah begin mathsize 14px style 2 square root of 2 end style.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 12 Maret 2021

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