Diketahui . Jika F"(x) habis dibagi x - 1 dan kurva y = F(x) mempunyai titik ekstrem lokal, maka...

Diketahui $begin mathsize 12px style F open parentheses x close parentheses equals b x cubed minus 3 open parentheses 1 plus a close parentheses x squared minus 3 x end style$ . Jika F"(x) habis dibagi x - 1 dan kurva y = F(x) mempunyai titik ekstrem lokal, maka...

$begin mathsize 12px style 0 less or equal than a less or equal than 1 end style$
$begin mathsize 12px style a less or equal than 0 space a t a u space a greater or equal than 1 end style$
$begin mathsize 12px style negative 1 less than a less than 0 end style$
$begin mathsize 12px style a less than negative 1 space a t a u space a greater than 0 end style$
$begin mathsize 12px style a less or equal than negative 2 space a t a u space a greater or equal than negative 1 end style$

Jawaban:

$begin mathsize 12px style F apostrophe open parentheses x close parentheses equals 3 b x squared minus 6 open parentheses 1 plus a close parentheses x minus 3 F " open parentheses x close parentheses equals 6 b x minus 6 open parentheses 1 plus a close parentheses end style$

F"(x) habis dibagi x - 1 artinya F"(1)=0, sehingga:

6b(1) - 6(1 + a) = 0

6b = 6a + 6 (kedua ruas dibagi dengan 6, maka)

b = a + 1

dengan mensubstitusikan b = a + 1 ke persamaan fungsi awal, maka diperoleh

$begin mathsize 12px style F open parentheses x close parentheses equals b x cubed minus 3 open parentheses 1 plus a close parentheses x squared minus 3 x F open parentheses x close parentheses equals open parentheses a plus 1 close parentheses x cubed minus 3 open parentheses 1 plus a close parentheses x squared minus 3 x end style$

Kurva y = F(x) mempunyai titik ekstrem lokal jika turunan pertamanya hanya memiliki paling banyak 1 akar, yaitu:

$begin mathsize 12px style F apostrophe open parentheses x close parentheses equals 0 3 open parentheses a plus 1 close parentheses x squared minus 6 open parentheses 1 plus a close parentheses x minus 3 equals 0 end style$

Sehingga akar dari turunan pertama F(x) paling banyak 1, maka $begin mathsize 12px style open parentheses D greater or equal than 0 close parentheses end style$ :

$size 12px D size 12px equals size 12px b to the power of size 12px 2 size 12px minus size 12px 4 size 12px a size 12px c open square brackets size 12px 6 open parentheses size 12px a size 12px plus size 12px 1 close parentheses close square brackets to the power of size 12px 2 size 12px minus size 12px 4 open square brackets size 12px 3 open parentheses size 12px 1 size 12px plus size 12px a close parentheses open parentheses size 12px minus size 12px 3 close parentheses close square brackets size 12px greater or equal than size 12px 0 size 12px 6 to the power of size 12px 2 open parentheses size 12px a size 12px plus size 12px 1 close parentheses to the power of size 12px 2 size 12px plus size 12px 36 open parentheses size 12px a size 12px plus size 12px 1 close parentheses size 12px greater or equal than size 12px 0 open parentheses size 12px 36 open parentheses size 12px a size 12px plus size 12px 1 close parentheses size 12px plus size 12px 36 close parentheses open parentheses size 12px a size 12px plus size 12px 1 close parentheses size 12px greater or equal than size 12px 0 size 12px 36 open parentheses size 12px a size 12px plus size 12px 1 close parentheses size 12px plus size 12px 36 size 12px equals size 12px 0 size 12px space size 12px a size 12px t size 12px a size 12px u size 12px space size 12px a size 12px plus size 12px 1 size 12px equals size 12px 0 size 12px a size 12px plus size 12px 1 size 12px equals size 12px minus size 12px 1 size 12px a size 12px equals size 12px minus size 12px 2 size 12px space size 12px a size 12px t size 12px a size 12px u size 12px space size 12px a size 12px space size 12px equals size 12px space size 12px minus size 12px 1 size 12px space size 12px space size 12px space open parentheses size 12px pembuat size 12px space size 12px nol close parentheses$

Dengan membuat garis bilangan, maka:

Jadi, kurva y = F(x) mempunyai titik ekstrem lokal, jika $begin mathsize 12px style a less or equal than negative 2 space a t a u space a greater or equal than negative 1 end style$

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