1: Ta có: \(-1<=\sin\left(2x+\frac{\pi}{4}\right)\le1\)
=>\(-3\le3\cdot\sin\left(2x+\frac{\pi}{4}\right)\le3\)
=>\(-3-1\le3\cdot\sin\left(2x+\frac{\pi}{4}\right)-1\le3-1\)
=>-4<=y<=2
=>Tập giá trị là T=[-4;2]
\(y_{\min}=-4\) khi \(\sin\left(2x+\frac{\pi}{4}\right)=-1\)
=>\(2x+\frac{\pi}{4}=-\frac{\pi}{2}+k2\pi\)
=>\(2x=-\frac34\pi+k2\pi\)
=>\(x=-\frac38\pi+k\pi\)
2: \(0\le cos^2x\le1\)
=>\(0\ge-5\cdot cos^2x\ge-5\)
=>\(0+3\ge-5\cdot cos^2x+3\ge-5+3\)
=>3>=y>=-2
=>Tập giá trị là T=[-2;3]
\(y_{\max}=3\) khi \(cos^2x=1\)
=>\(\sin^2x=0\)
=>sin x=0
=>\(x=k\pi\)
\(y_{\min}=-2\) khi \(cos^2x=0\)
=>cosx=0
=>\(x=\frac{k\pi}{2}\)
3: \(-1\le cosx\le1\)
=>\(-3\le3\cdot cosx\le3\)
=>\(-3+4\le3\cdot cosx+4\le3+4\)
=>\(1\le3\cdot cosx+4\le7\)
=>\(\frac51\ge\frac{5}{3\cdot cosx+4}\ge\frac57\)
=>\(\frac57\le y\le5\)
=>Tập giá trị là \(T=\left\lbrack\frac57;5\right\rbrack\)
\(y_{\min}=\frac57\) khi cosx=1
=>\(x=k2\pi\)
\(y_{\max}=5\) khi cosx=-1
=>\(x=\pi+k2\pi\)
4: \(y=\sin^2x-4\cdot\sin x+8\)
\(=\sin^2x-4\cdot\sin x+4+4\)
\(=\left(\sin x-2\right)^2+4\)
Ta có: \(-1\le\sin x\le1\)
=>\(-1-2\le\sin x-2\le1-2\)
=>\(-3\le\sin x-2\le-1\)
=>\(1\le\left(\sin x-2\right)^2\le9\)
=>\(5\le\left(\sin x-2\right)^2+4\le13\)
=>5<=y<=13
=>Tập giá trị là T=[5;13]
\(y_{\min}=5\) khi sin x-2=-1
=>sin x=1
=>\(x=\frac{\pi}{2}+k2\pi\)
\(y_{\max}\) =13 khi sin x-2=-3
=>sin x=-1
=>\(x=-\frac{\pi}{2}+k2\pi\)
