a, Đặt \(t=cos3x\left(t\in\left[-1;1\right]\right)\)
\(y=9-sin^23x-\sqrt{2}cos3x\)
\(=cos^23x-\sqrt{2}cos3x+8\)
\(\Leftrightarrow y=f\left(t\right)=t^2-\sqrt{2}t+8\)
\(\Rightarrow minf\left(t\right)\le y\le maxf\left(x\right)\)
\(\Rightarrow min\left\{f\left(-1\right);f\left(1\right);f\left(\dfrac{\sqrt{2}}{2}\right)\right\}\le y\le max\left\{f\left(-1\right);f\left(1\right);f\left(\dfrac{\sqrt{2}}{2}\right)\right\}\)
\(\Rightarrow\dfrac{15}{2}\le y\le9+\sqrt{2}\)
\(\Rightarrow y_{max}=9+\sqrt{2}\)
b, Đặt \(t=sin3x\left(t\in\left[-1;1\right]\right)\)
\(y=3sin3x-8cos^23x+4\)
\(=3sin3x+8-8cos^23x-4\)
\(=8sin^23x+3sin3x-4\)
\(\Leftrightarrow y=f\left(t\right)=8t^2+3t-4\)
\(\Rightarrow minf\left(x\right)\le y\le maxf\left(t\right)\)
\(\Rightarrow min\left\{f\left(-1\right);f\left(1\right);f\left(-\dfrac{3}{16}\right)\right\}\le y\le max\left\{f\left(-1\right);f\left(1\right);f\left(-\dfrac{3}{16}\right)\right\}\)
\(\Rightarrow-\dfrac{137}{32}\le y\le7\)
\(\Rightarrow y_{max}=7\)
