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Misalkan , maka nilai maksimum dari 2(f(x))² adalah ....

Misalkan begin mathsize 14px style straight f open parentheses straight x close parentheses equals open parentheses square root of 7 plus 2 close parentheses cos invisible function application straight x plus open parentheses square root of 7 minus 2 close parentheses sin invisible function application straight x end style, maka nilai maksimum dari 2(f(x))² adalah ....

  1. 4

  2. 22

  3. 44

  4. 88

  5. 100

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R. Rohmat

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

jawaban yang tepat adalah C.

Pembahasan

Perhatikan bahwa : Perhatikan bahwa dengan . Perhatikan bentuk . Karena dan b = 3, maka Sehingga Maka Ingat bahwa Sehingga nilai maksimum dari 2(f(x))² adalah 44. Jadi, jawaban yang tepat adalah C.

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 7 plus 2 close parentheses cos invisible function application straight x plus open parentheses square root of 7 minus 2 close parentheses sin 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 7 plus 2 close parentheses cos invisible function application straight x plus open parentheses square root of 7 minus 2 close parentheses sin 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 7 plus 2 close parentheses cos invisible function application straight x close parentheses squared plus 2 open parentheses square root of 7 plus 2 close parentheses cos invisible function application straight x open parentheses square root of 7 minus 2 close parentheses sin invisible function application straight x plus open parentheses open parentheses square root of 7 minus 2 close parentheses sin 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 7 plus 2 close parentheses squared cos squared invisible function application straight x plus 2 cos invisible function application xsin invisible function application straight x open parentheses square root of 7 plus 2 close parentheses open parentheses square root of 7 minus 2 close parentheses plus open parentheses square root of 7 minus 2 close parentheses squared sin 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 7 plus 2 close parentheses squared cos squared invisible function application straight x plus 2 cos invisible function application xsin invisible function application straight x open parentheses 7 minus 4 close parentheses plus open parentheses square root of 7 minus 2 close parentheses squared sin 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 7 plus 2 close parentheses squared cos squared invisible function application straight x plus 2 cos invisible function application xsin invisible function application straight x open parentheses 3 close parentheses plus open parentheses square root of 7 minus 2 close parentheses squared sin 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 7 plus 2 close parentheses squared cos squared invisible function application straight x plus 3 times 2 cos invisible function application xsin invisible function application straight x plus open parentheses square root of 7 minus 2 close parentheses squared sin 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 7 plus 2 close parentheses squared cos squared invisible function application straight x plus 3 sin invisible function application 2 straight x plus open parentheses square root of 7 minus 2 close parentheses squared sin 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 7 plus 2 close parentheses squared cos squared invisible function application straight x plus open parentheses square root of 7 minus 2 close parentheses squared sin squared invisible function application straight x plus 3 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 square root of 7 plus 2 close parentheses squared cos squared invisible function application straight x plus open parentheses square root of 7 minus 2 close parentheses squared sin squared invisible function application straight x plus 3 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 7 plus 4 square root of 7 plus 4 close parentheses cos squared invisible function application straight x plus open parentheses 7 minus 4 square root of 7 plus 4 close parentheses sin squared invisible function application straight x plus 3 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 11 plus 4 square root of 7 close parentheses cos squared invisible function application straight x plus open parentheses 11 minus 4 square root of 7 close parentheses sin squared invisible function application straight x plus 3 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 11 cos squared invisible function application straight x plus 4 square root of 7 cos squared invisible function application straight x plus 11 sin squared invisible function application straight x minus 4 square root of 7 sin squared invisible function application straight x plus 3 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 11 cos squared invisible function application straight x plus 11 sin squared invisible function application straight x plus 4 square root of 7 cos squared invisible function application straight x minus 4 square root of 7 sin squared invisible function application straight x plus 3 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 11 open parentheses cos squared invisible function application straight x plus sin squared invisible function application straight x close parentheses plus 4 square root of 7 open parentheses cos squared invisible function application straight x minus sin squared invisible function application straight x close parentheses plus 3 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 11 open parentheses 1 close parentheses plus 4 square root of 7 cos invisible function application 2 straight x plus 3 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 11 plus 4 square root of 7 cos invisible function application 2 straight x plus 3 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 4 square root of 7 cos invisible function application 2 straight x plus 3 sin invisible function application 2 straight x end style. Karena begin mathsize 14px style straight a equals 4 square root of 7 end style dan b = 3, 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 4 square root of 7 close parentheses squared plus 3 squared end root end cell row blank equals cell square root of 112 plus 9 end root end cell row blank equals cell square root of 121 end cell row blank equals 11 end table end style  


Sehingga begin mathsize 14px style 4 square root of 7 cos invisible function application 2 straight x plus 3 sin invisible function application 2 straight x equals 11 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses end style 

Maka

begin mathsize 14px style open parentheses straight f open parentheses straight x close parentheses close parentheses squared equals 11 plus 4 square root of 7 cos invisible function application 2 straight x plus 3 sin invisible function application 2 straight x open parentheses straight f open parentheses straight x close parentheses close parentheses squared equals 11 plus 11 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses 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 11 end cell less or equal than cell 11 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses less or equal than 11 end cell row cell 11 minus 11 end cell less or equal than cell 11 plus 11 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses less or equal than 11 plus 11 end cell row 0 less or equal than cell 11 plus 11 cos invisible function application open parentheses 2 straight x minus straight alpha close parentheses less or equal than 22 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 22 end cell row 0 less or equal than cell 2 open parentheses straight f open parentheses straight x close parentheses close parentheses squared less or equal than 44 end cell end table end style 


Sehingga nilai maksimum dari 2(f(x))² adalah 44.

Jadi, jawaban yang tepat adalah C.

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