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Đặt \(cos2x=t\in\left[-1;1\right]\)
\(\Rightarrow y=f\left(t\right)=t^2+2t\)
Xét hàm \(f\left(t\right)=t^2+2t\) trên \(\left[-1;1\right]\)
\(-\dfrac{b}{2a}=-1\in\left[-1;1\right]\)
\(f\left(-1\right)=-1\) ; \(f\left(1\right)=3\)
\(\Rightarrow y_{min}=-1\) khi \(cos2x=-1\Rightarrow x=\dfrac{\pi}{2}+k\pi\)
\(y_{max}=3\) khi \(cos2x=1\Rightarrow x=k\pi\)
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\(y=\sqrt{3}cosx-sinx=2\left(\dfrac{\sqrt{3}}{2}cosx-\dfrac{1}{2}sinx\right)=2cos\left(x+\dfrac{\pi}{6}\right)\)
Vì \(cos\left(x+\dfrac{\pi}{6}\right)\in\left[-1;1\right]\Rightarrow y=\sqrt{3}cosx-sinx\in\left[-2;2\right]\)
\(\Rightarrow y_{min}=-2\Leftrightarrow cos\left(x+\dfrac{\pi}{6}\right)=-1\Leftrightarrow x+\dfrac{\pi}{6}=\pi+k2\pi\Leftrightarrow x=\dfrac{5\pi}{6}+k2\pi\)
\(y_{max}=2\Leftrightarrow cos\left(x+\dfrac{\pi}{6}\right)=1\Leftrightarrow x+\dfrac{\pi}{6}=k2\pi\Leftrightarrow x=-\dfrac{\pi}{6}+k2\pi\)
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Em không thấy đáp án giống như trên lời giải, có thể giúp em làm cách khác không ạ?
Bạn Phúc hơi nhầm 1 xíu
\(y=4sinx\left(\dfrac{1}{2}cosx-\dfrac{\sqrt{3}}{2}sinx\right)=2sinx.cosx-2\sqrt{3}sin^2x\)
\(=sin2x-\sqrt{3}\left(1-cos2x\right)=sin2x+\sqrt{3}cos2x-\sqrt{3}\)
\(=2\left(\dfrac{\sqrt{3}}{2}cos2x+\dfrac{1}{2}sin2x\right)-\sqrt{3}\)
\(=2cos\left(2x-\dfrac{\pi}{6}\right)-\sqrt{3}\)
\(\Rightarrow y_{min}=-2-\sqrt{3}\) ; \(y_{max}=2-\sqrt{3}\)
Đáp án mà đề đưa ra như bên dưới đều sai cả.
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(sinx=t\left(t\in\left[-1;1\right]\right)\)
\(y=\left|sinx+cos2x\right|=\left|2sin^2x-sinx-1\right|\)
\(\Leftrightarrow y=\left|f\left(t\right)\right|=\left|2t^2-t-1\right|\)
\(f\left(-1\right)=2\Rightarrow y=2\)
\(f\left(1\right)=0\Rightarrow y=0\)
\(f\left(\dfrac{1}{4}\right)=-\dfrac{9}{8}\Rightarrow y=\dfrac{9}{8}\)
\(\Rightarrow y_{min}=0;y_{max}=2\)
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21.
a) `2sin(x-30^@)-1=0`
`<=>sin(x-30^@)=1/2`
`<=> sin(x-30^@)=sin30^@`
`<=>[(x-30^@=30^@+k360^@),(x-30^@=180^@-30^@+k360^@):}`
`<=> [(x=60^@+k360^@),(x=180^@+k360^@):}`
b) `5sin^2x+3cosx+3=0`
`<=>5(1-cos^2x)+3cosx+3=0`
`<=>-5cos^2x+3cosx+8=0`
`<=>(cosx+1)(cosx=8/5)=0`
`<=>[(cosx=-1),(cosx=8/5\ (VN)):}`
`<=>x=180^@+k360^@`
22.
`-1<=sin2x<=1`
`<=>2<=3+sin2x<=4`
`=> y_(min)=2 ; y_(max)=4`