\(y=8-3sin^23x+6sin6x\)
\(=8+\dfrac{3}{2}\left(1-2sin^23x\right)+6sin6x-\dfrac{3}{2}\)
\(=\dfrac{3}{2}cos6x+6sin6x+\dfrac{13}{2}\)
\(=\dfrac{3\sqrt{17}}{2}\left(\dfrac{1}{\sqrt{17}}cos6x+\dfrac{4}{\sqrt{17}}sin6x\right)+\dfrac{13}{2}\)
\(=\dfrac{3\sqrt{17}}{2}cos\left(6x-arccos\dfrac{1}{\sqrt{17}}\right)+\dfrac{13}{2}\)
\(\le-\dfrac{3\sqrt{17}}{2}+\dfrac{13}{2}=\dfrac{13-3\sqrt{17}}{2}\)
\(y_{max}=\dfrac{13-3\sqrt{17}}{2}\Leftrightarrow cos\left(6x-arccos\dfrac{1}{\sqrt{17}}\right)=1\Leftrightarrow x=\dfrac{1}{6}arccos\dfrac{1}{\sqrt{17}}+\dfrac{k\pi}{3}\)
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