5: \(y=cos2x+cosx+1\)
\(=2\cdot cos^2x-1+cosx+1\)
\(=2\cdot cos^2x+cosx=2\cdot\left(cos^2x+cosx\right)\)
\(=2\cdot\left(cos^2x+cosx+\frac14\right)-\frac12\)
\(=2\cdot\left(cosx+\frac12\right)^2-\frac12\)
Ta có: \(-1\le cosx\le1\)
=>\(-1+\frac12\le cosx+\frac12\le1+\frac12\)
=>\(-\frac12\le cosx+\frac12\le\frac32\)
=>\(0\le\left(cosx+\frac12\right)^2\le\frac94\)
=>\(0\le2\cdot\left(cosx+\frac12\right)^2\le\frac94\cdot2=\frac92\)
=>\(0-\frac12\le2\cdot\left(cosx+\frac12\right)^2-\frac12\le\frac92-\frac12\)
=>\(-\frac12\le y\le4\)
=>Tập giá trị là T=[-1/2;4]
\(y_{\min}=-\frac12\) khi \(cosx+\frac12=0\)
=>\(cosx=-\frac12\)
=>\(x=\pm\frac23\pi+k2\pi\)
\(y_{\max}=4\) khi \(cosx+\frac12=\frac32\)
=>cosx=1
=>\(x=k2\pi\)
6: \(y=-2\cdot\sin^2x+cosx-5\)
\(=-2\cdot\left(1-cos^2x\right)+cosx-5\)
\(=-2+2\cdot cos^2x+cosx-5=2\cdot cos^2x+cosx-7\)
\(=2\cdot\left(cos^2x+\frac12\cdot cosx-\frac72\right)\)
\(=2\cdot\left(cos^2x+2\cdot cosx\cdot\frac14+\frac{1}{16}-\frac{57}{16}\right)\)
\(=2\cdot\left\lbrack\left(cosx+\frac14\right)^2-\frac{57}{16}\right\rbrack\)
\(=2\cdot\left(cosx+\frac14\right)^2-\frac{57}{8}\)
Ta có: \(-1<=cosx\le1\)
=>\(-1+\frac14\le cosx+\frac14\le1+\frac14\)
=>\(-\frac34\le cosx+\frac14\le\frac54\)
=>\(0\le\left(cosx+\frac14\right)^2\le\frac{25}{16}\)
=>\(0\le2\cdot\left(cosx+\frac14\right)^2\le\frac{25}{8}\)
=>\(0-\frac{57}{8}\le2\cdot\left(cosx+\frac14\right)^2-\frac{57}{8}\le\frac{25}{8}-\frac{57}{8}\)
=>\(-\frac{57}{8}\le y\le-\frac{32}{8}=-4\)
=>Tập giá trị là \(T=\left\lbrack-\frac{57}{8};-4\right\rbrack\)
\(y_{\min}=-\frac{57}{8}\) khi \(cosx+\frac14=0\)
=>\(cosx=-\frac14\)
=>\(x=\pm arccos\left(-\frac14\right)+k2\pi\)
\(y_{\min}=-4\) khi \(cosx+\frac14=\frac54\)
=>cosx=1
=>\(x=k2\pi\)
7: \(y=\sin x+cosx-2\)
\(=\sqrt2\cdot\sin\left(x+\frac{\pi}{4}\right)-2\)
Ta có: \(-1\le\sin\left(x+\frac{\pi}{4}\right)\le1\)
=>\(-\sqrt2\le\sqrt2\cdot\sin\left(x+\frac{\pi}{4}\right)\le\sqrt2\)
=>\(-\sqrt2-2\le\sqrt2\cdot\sin\left(x+\frac{\pi}{4}\right)-2\le\sqrt2-2\)
=>Tập giá trị là: \(T=\left\lbrack-\sqrt2-2;\sqrt2-2\right\rbrack\)
\(y_{\min}=-\sqrt2-2\) khi \(\sin\left(x+\frac{\pi}{4}\right)=-1\)
=>\(x+\frac{\pi}{4}=-\frac{\pi}{2}+k2\pi\)
=>\(x=-\frac34\pi+k2\pi\)
\(y_{\max}=\sqrt2-2\) khi \(\sin\left(x+\frac{\pi}{4}\right)=1\)
=>\(x+\frac{\pi}{4}=\frac{\pi}{2}+k2\pi\)
=>\(x=\frac{\pi}{4}+k2\pi\)
8: \(y=3\cdot\sin2x-4\cdot cos2x+1\)
\(=5\cdot\left\lbrack\frac35\cdot\sin2x-\frac45\cdot cos2x\right\rbrack+1\)
\(=5\cdot\left\lbrack\sin2x\cdot cos\alpha-cos2x\cdot\sin\alpha\right\rbrack+1\) với cosα=3/5; sin α=4/5
\(=5\cdot\sin\left(2x-\alpha\right)+1\)
Ta có: \(-1<=\sin\left(2x-\alpha\right)\le1\)
=>\(-5\le5\cdot\sin\left(2x-\alpha\right)\le5\)
=>\(-5+1\le5\cdot\sin\left(2x-\alpha\right)+1\le5+1\)
=>\(-4\le5\cdot\sin\left(2x-\alpha\right)+1\le6\)
=>-4<=y<=6
=>Tập giá trị là T=[-4;6]
\(y_{\min}=-4\) khi sin(2x-α)=-1
=>\(2x-\alpha=-\frac{\pi}{2}+k2\pi\)
=>\(2x=\alpha-\frac{\pi}{2}+k2\pi\)
=>\(x=\frac{\alpha}{2}-\frac{\pi}{4}+k\pi\)
\(y_{\max}=6\) khi sin(2x-α)=1
=>\(2x-\alpha=\frac{\pi}{2}+k2\pi\)
=>\(2x=\alpha+\frac{\pi}{2}+k2\pi\)
=>\(x=\frac{\alpha}{2}+\frac{\pi}{4}+k\pi\)
