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Misalkan . Nilai maksimum untuk (f(x))² dengan x merupakanbilangan real adalah ....

Misalkan begin mathsize 14px style straight f open parentheses straight x close parentheses equals open parentheses square root of 2 plus 1 close parentheses sin invisible function application straight x plus open parentheses square root of 2 minus 1 close parentheses cos invisible function application straight x end style. Nilai maksimum untuk (f(x))² dengan x merupakan bilangan real adalah ....

  1. –6

  2. –3

  3. 0

  4. 3

  5. 6

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R. Rohmat

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

jawaban yang tepat adalah E.

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Pembahasan

Perhatikan bahwa Perhatikan bahwa : dengan Perhatikan bentuk . Karena dan b = 1, maka : Sehingga . Maka Ingat bahwa Sehingga nilai maksimum untuk (f(x))² adalah 6. Jadi, jawaban yang tepat adalah E.

Perhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses straight x close parentheses end cell equals cell open parentheses square root of 2 plus 1 close parentheses sin invisible function application straight x plus open parentheses square root of 2 minus 1 close parentheses cos invisible function application straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses open parentheses square root of 2 plus 1 close parentheses sin invisible function application straight x plus open parentheses square root of 2 minus 1 close parentheses cos invisible function application straight x close parentheses squared end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses open parentheses square root of 2 plus 1 close parentheses sin invisible function application straight x close parentheses squared plus 2 open parentheses square root of 2 plus 1 close parentheses sin invisible function application straight x open parentheses square root of 2 minus 1 close parentheses cos invisible function application straight x plus open parentheses open parentheses square root of 2 minus 1 close parentheses cos invisible function application straight x close parentheses squared end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses square root of 2 plus 1 close parentheses squared sin squared invisible function application straight x plus 2 sin invisible function application xcos invisible function application straight x open parentheses square root of 2 plus 1 close parentheses open parentheses square root of 2 minus 1 close parentheses plus open parentheses square root of 2 minus 1 close parentheses squared cos squared invisible function application straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses square root of 2 plus 1 close parentheses squared sin squared invisible function application straight x plus 2 sin invisible function application xcos invisible function application straight x open parentheses 2 minus 1 close parentheses plus open parentheses square root of 2 minus 1 close parentheses squared cos squared invisible function application straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses square root of 2 plus 1 close parentheses squared sin squared invisible function application straight x plus 2 sin invisible function application xcos invisible function application straight x open parentheses 1 close parentheses plus open parentheses square root of 2 minus 1 close parentheses squared cos squared invisible function application straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses square root of 2 plus 1 close parentheses squared sin squared invisible function application straight x plus 2 sin invisible function application xcos invisible function application straight x plus open parentheses square root of 2 minus 1 close parentheses squared cos squared invisible function application straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses square root of 2 plus 1 close parentheses squared sin squared invisible function application straight x plus sin invisible function application 2 straight x plus open parentheses square root of 2 minus 1 close parentheses squared cos squared invisible function application straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses square root of 2 plus 1 close parentheses squared sin squared invisible function application straight x plus open parentheses square root of 2 minus 1 close parentheses squared cos squared invisible function application straight x plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses 2 plus 2 square root of 2 plus 1 close parentheses sin squared invisible function application straight x plus open parentheses 2 minus 2 square root of 2 plus 1 close parentheses cos squared invisible function application straight x plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell open parentheses 3 plus 2 square root of 2 close parentheses sin squared invisible function application straight x plus open parentheses 3 minus 2 square root of 2 close parentheses cos squared invisible function application straight x plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell 3 sin squared invisible function application straight x plus 2 square root of 2 sin squared invisible function application straight x plus 3 cos squared invisible function application straight x minus 2 square root of 2 cos squared invisible function application straight x plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell 3 sin squared invisible function application straight x plus 3 cos squared invisible function application straight x minus 2 square root of 2 cos squared invisible function application straight x plus 2 square root of 2 sin squared invisible function application straight x plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell 3 open parentheses sin squared invisible function application straight x plus cos squared invisible function application straight x close parentheses minus 2 square root of 2 open parentheses cos squared invisible function application straight x minus sin squared invisible function application straight x close parentheses plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell 3 open parentheses 1 close parentheses minus 2 square root of 2 cos invisible function application 2 straight x plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell 3 minus 2 square root of 2 cos invisible function application 2 straight x plus sin invisible function application 2 straight x end cell end table end style 


Perhatikan bahwa :

begin mathsize 14px style straight a space cos invisible function application straight x plus straight b space sin invisible function application straight x equals straight k space cos invisible function application open parentheses straight x minus straight alpha close parentheses end style 

dengan begin mathsize 14px style straight k equals square root of straight a squared plus straight b squared end root end style 


Perhatikan bentuk begin mathsize 14px style negative 2 square root of 2 cos invisible function application 2 straight x plus sin invisible function application 2 straight x end style. Karena begin mathsize 14px style straight a equals negative 2 square root of 2 end style dan b = 1, maka :

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row straight k equals cell square root of open parentheses negative 2 square root of 2 close parentheses squared plus 1 squared end root end cell row blank equals cell square root of 8 plus 1 end root end cell row blank equals cell square root of 9 end cell row blank equals 3 end table end style 

Sehingga begin mathsize 14px style negative 2 square root of 2 cos invisible function application 2 straight x plus sin invisible function application 2 straight x equals 3 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses end style.


Maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell 3 minus 2 square root of 2 cos invisible function application 2 straight x plus sin invisible function application 2 straight x end cell row cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared end cell equals cell 3 plus 3 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses end cell end table end style 


Ingat bahwa 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 1 end cell less or equal than cell cos invisible function application straight x less or equal than 1 end cell row cell negative 1 end cell less or equal than cell cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses less or equal than 1 end cell row cell negative 3 end cell less or equal than cell 3 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses less or equal than 3 end cell row cell 3 minus 3 end cell less or equal than cell 3 plus 3 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses less or equal than 3 plus 3 end cell row 0 less or equal than cell 3 plus 3 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses less or equal than 6 end cell row 0 less or equal than cell open parentheses straight f open parentheses straight x close parentheses close parentheses squared less or equal than 6 end cell end table end style 

Sehingga nilai maksimum untuk (f(x))² adalah 6.

Jadi, jawaban yang tepat adalah E.

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