1.
\(-1\le sin\left(x-\dfrac{\pi}{2}\right)\le1\Rightarrow1\le y\le5\)
\(y_{min}=1\) khi \(sin\left(x-\dfrac{\pi}{2}\right)=-1\)
\(y_{max}=5\) khi \(sin\left(x-\dfrac{\pi}{2}\right)=1\)
2.
\(-1\le cos2x\le1\Rightarrow\dfrac{5}{2}\le y\le\dfrac{7}{2}\)
\(y_{min}=\dfrac{5}{2}\) khi \(cos2x=1\)
\(y_{max}=\dfrac{7}{2}\) khi \(cos2x=-1\)
3.
\(0\le cos^2\left(2x+\dfrac{\pi}{3}\right)\le1\Rightarrow-2\le y\le-1\)
\(y_{min}=-2\) khi \(cos\left(2x+\dfrac{\pi}{3}\right)=\pm1\)
\(y_{max}=-1\) khi \(cos\left(2x+\dfrac{\pi}{3}\right)=0\)
4.
\(-1\le cos\left(4x^2\right)\le1\Rightarrow-2\le y\le\sqrt{2}-2\)
\(y_{min}=-1\) khi \(cos\left(4x^2\right)=-1\)
\(y_{max}=\sqrt{2}-2\) khi \(cos\left(4x^2\right)=1\)
5.
\(0\le\sqrt{sinx}\le1\Rightarrow3\le y\le5\)
\(y_{min}=3\) khi \(sinx=0\)
\(y_{max}=5\) khi \(sinx=1\)
6.
\(-1\le cos\sqrt{x+\dfrac{\pi}{4}}\le1\Rightarrow-5\le y\le5\)
\(y_{min}=-5\) khi \(cos\sqrt{x+\dfrac{\pi}{4}}=-1\)
\(y_{max}=5\) khi \(cos\sqrt{x+\dfrac{\pi}{4}}=1\)
7.
\(y=\left(1-sinx\right)\left(3-sinx\right)\)
Do \(sinx\le1\Rightarrow\left\{{}\begin{matrix}1-sinx\ge0\\3-sinx>0\end{matrix}\right.\) \(\Rightarrow y\ge0\)
\(y_{min}=0\) khi \(sinx=1\)
\(y=sin^2x-4sinx-5+8=\left(sinx+1\right)\left(sinx-5\right)+8\)
Do \(-1\le sinx\le1\Rightarrow\left(sinx+1\right)\left(sinx-5\right)\le0\)
\(\Rightarrow y\le8\)
\(y_{max}=8\) khi \(sinx=-1\)
8.
\(0\le cos^23x\le1\Rightarrow2\le y\le3\)
\(y_{min}=2\) khi \(cos^23x=1\)
\(y_{max}=3\) khi \(cos3x=0\)
