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SMK Teknologi dan Rekayasa

Soal 118 Nilai Maksimum-Minimum Fungsi Aljabar


Nilai\ maksimum\ fungsi\ f(x)=2x^{3}-24x^{2}+23\\pada\ interval\ -1\leq x\leq 1\ adalah\ ....\\a.\ \ 10\ \ \ \ \ \ \ \ \ \ b.\ \ 23\ \ \ \ \ \ \ \ \ \ c.\ \ 39\ \ \ \ \ \ \ \ \ \ d.\ \ 41\ \ \ \ \ \ \ \ \ \ e.\ \ 55
Jawaban: b

Nilai\ maksimum\ (atau\ minimum)\ fungsi\ dalam\ interval\ tertutup\ terjadi\\\mathbf{pada\ stasioner\ atau\ pada\ batas\ interval}.
f(x)=2x^{3}-24x^{2}+23\\f{}'(x)=2.3x^{3-1}-24.2x^{2-1}+0=6x^{2}-48x=6x(x-8)

Stasioner fungsi:
\begin{array}{rcl}f{}'(x)&=&0\\6x(x-8)&=&0\\6x=0&atau&x-8=0\\x=0&atau&x=8\end{array}

Nilai\ ekstrim\ fungsi:
f(0)=2.0^{3}-24.0^{2}+23=2.0-24.0+23=0-0+23=23
f(8)=2.8^{3}-24.8^{2}+23=2.512-24.64+23=1024-1536+23=-489
f(-1)=2.(-1)^{3}-24.(-1)^{2}+23=2.(-1)-24.1+23=-2-24+23=-3
f(1)=2.(1)^{3}-24.(1)^{2}+23=2.1-24.1+23=2-24+23=1
Nilai\ maksimumnya\ adalah\ 23.

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