\(y=2\left(\frac{1}{2}sin2x+\frac{\sqrt{3}}{2}cos2x\right)+1=2sin\left(2x+\frac{\pi}{3}\right)+1\)

Do \(-1\le sin\left(2x+\frac{\pi}{3}\right)\le1\)

\(\Rightarrow-1\le y\le3\)

a.

\(-1\le sinx\le1\Rightarrow-7\le y\le-3\)

\(y_{min}=-7\) khi \(sinx=-1\)

\(y_{max}=-3\) khi \(sinx=1\)

b.

\(-1\le cos\left(x+\frac{\pi}{3}\right)\le1\Rightarrow1\le y\le5\)

\(y_{min}=1\) khi \(cos\left(x+\frac{\pi}{3}\right)=-1\)

\(y_{max}=5\) khi \(cos\left(x+\frac{\pi}{3}\right)=1\)

c.

\(0\le1-cosx\le2\Rightarrow-5\le y\le3\sqrt{2}-5\)

\(y_{min}=-5\) khi \(cosx=1\)

\(y_{max}=3\sqrt{2}-5\) khi \(cosx=-1\)

d.

ĐKXĐ: \(0\le sinx\Rightarrow0\le sinx\le1\Rightarrow1\le y\le3\)

\(y_{min}=1\) khi \(sinx=0\)

\(y_{max}=3\) khi \(sinx=1\)

a/ \(y=sin2x+\left(\sqrt{3}+1\right)cos2x+sin^2x-cos^2x-1\)

\(=sin2x+\sqrt{3}cos2x-1=2sin\left(2x+\frac{\pi}{3}\right)-1\)

Do \(-1\le sin\left(2x+\frac{\pi}{3}\right)\le1\Rightarrow-3\le y\le1\)

b/ \(y=2sin^2x-2cos^2x-3sinx.cosx-1\)

\(=-2cos2x-\frac{3}{2}sin2x-1=-\frac{5}{2}\left(\frac{3}{5}sinx+\frac{4}{5}cosx\right)-1\)

\(=-\frac{5}{2}sin\left(x+a\right)-1\Rightarrow-\frac{7}{2}\le y\le\frac{3}{2}\)

c/ \(y=1-sin2x+2cos2x+\frac{3}{2}sin2x=\frac{1}{2}sin2x+2cos2x+1\)

\(=\frac{\sqrt{17}}{2}\left(\frac{1}{\sqrt{17}}sin2x+\frac{4}{\sqrt{17}}cos2x\right)+1=\frac{\sqrt{17}}{2}sin\left(2x+a\right)+1\)

\(\Rightarrow-\frac{\sqrt{17}}{2}+1\le y\le\frac{\sqrt{17}}{2}+1\)