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SIMAK UI 2010

Solusi SIMAK UI 2010 Kode 509 [IPA] – Nomor 1


Jumlah\ nilai\ terbesar\ dan\ terkecil\ dari
\frac{x^2\ +\ 14x\ +\ 9}{x^2\ +\ 2x\ +\ 3}
untuk\ setiap\ nilai\ x\ real\ adalah\ ....
(A)\quad -3\\(B)\quad -2\\(C)\quad -1\\(D)\quad 1\\(E)\quad 2
Jawaban: C

Misalkan
f(x)=\frac{x^2+14x+9}{x^2+2x+3}=\frac{(x+7)^2-40}{(x+1)^2+2}
Syarat\ maksimum/minimum\ adalah\ f'(x)=0\ sehingga:
(2x+14)(x^2+2x+3)-(x^2+14x+9)(2x+2)=0\\(pembilang\ pada\ hasil\ diferensial=0)
(x+7)(x^2+2x+3)=(x^2+14x+9)(x+1)
x^3+9x^2+17x+21=x^3+x^2+15x^2+23x+9
6x^2+6x-12=0
x^2+x-2=0
(x-1)(x+2)=0
x=1\,\,\, atau\,\,\, x=-2
f(1)=\frac{(1+7)^2-40}{(1+1)^2+2}=\frac{64-40}{4+2}=\frac{24}{6}=4
f(-2)=\frac{(-2+7)^2-40}{(-2+1)^2+2}=\frac{25-40}{1+2}=\frac{-15}{3}=-5
Jadi\ jumlah\ nilai\ terbesar\ dan\ terkecil=4+(-5)=-1

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