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\(-1\le cos\left(\sqrt{x}+\dfrac{\pi}{4}\right)\le1\Rightarrow-5\le y\le5\)
\(y_{max}=5\) khi \(cos\left(\sqrt{x}+\dfrac{\pi}{4}\right)=1\)
\(y_{min}=-5\) khi \(cos\left(\sqrt{x}+\dfrac{\pi}{4}\right)=-1\)
1. Không dịch được đề
2.
\(-1\le cos2x\le1\Rightarrow1\le y\le3\)
3.
a. \(-2\le2sinx\le2\Rightarrow-1\le y\le3\)
\(y_{min}=-1\) khi \(sinx=-1\Rightarrow x=-\dfrac{\pi}{2}+k2\pi\)
\(y_{max}=3\) khi \(sinx=1\Rightarrow x=\dfrac{\pi}{2}+k2\pi\)
b.
\(0\le cos^2x\le1\Rightarrow-1\le y\le2\)
\(y_{min}=-1\) khi \(cos^2x=1\Rightarrow x=k\pi\)
\(y_{max}=2\) khi \(cosx=0\Rightarrow x=\dfrac{\pi}{2}+k\pi\)
4.
\(y=\left(tanx-1\right)^2+2\ge2\)
\(y_{min}=2\) khi \(tanx=1\Rightarrow x=\dfrac{\pi}{4}+k\pi\)
\(x\in\left[\dfrac{1}{4};\dfrac{3}{2}\right]\Rightarrow\pi x\in\left[\dfrac{\pi}{4};\dfrac{3\pi}{2}\right]\)
\(\Rightarrow cos\left(\pi x\right)\in\left[-1;\dfrac{\sqrt{2}}{2}\right]\)
\(y_{max}=\dfrac{\sqrt{2}}{2}\) khi \(x=\dfrac{1}{4}\)
\(y_{min}=-1\) khi \(x=1\)
\(y=-1-cos^2\left(2x+\dfrac{\pi}{3}\right)\)
\(=-\dfrac{3}{2}+\dfrac{1}{2}-cos^2\left(2x+\dfrac{\pi}{3}\right)\)
\(=-\dfrac{3}{2}-\dfrac{1}{2}\left[2cos^2\left(2x+\dfrac{\pi}{3}\right)-1\right]\)
\(=-\dfrac{3}{2}-\dfrac{1}{2}cos\left(4x+\dfrac{2\pi}{3}\right)\)
Vì \(cos\left(4x+\dfrac{2\pi}{3}\right)\in\left[-1;1\right]\)
\(\Rightarrow min=-\dfrac{3}{2}-\dfrac{1}{2}=-2\Leftrightarrow cos\left(4x+\dfrac{2\pi}{3}\right)=1\)
\(\Rightarrow max=-\dfrac{3}{2}+\dfrac{1}{2}=-1\Leftrightarrow cos\left(4x+\dfrac{2\pi}{3}\right)=-1\)
a)\(-1\le sinx\le1\)
\(\Leftrightarrow1\ge-sinx\ge-1\)
\(\Leftrightarrow4\ge3-sinx\ge2\) \(\Leftrightarrow16\ge\left(3-sinx\right)^2\ge4\)\(\Leftrightarrow17\ge\left(3-sinx\right)^2+1\ge5\)
\(\Leftrightarrow17\ge y\ge5\)
\(y_{min}=5\Leftrightarrow sinx=1\)\(\Leftrightarrow\)\(x=\dfrac{\pi}{2}+k2\pi\)\(\left(k\in Z\right)\)
\(y_{max}=17\Leftrightarrow\)\(sinx=-1\Leftrightarrow x=-\dfrac{\pi}{2}+k2\pi\)\(\left(k\in Z\right)\)
b)\(y=\left(sin^2x+cos^2x\right)^2-2.sinx^2cos^2x\)\(=1-\dfrac{1}{2}.sin^22x\)
Có \(0\le sin^22x\le1\)\(\Leftrightarrow0\ge-\dfrac{1}{2}.sin^22x\ge-\dfrac{1}{2}\)
\(\Leftrightarrow1\ge1-\dfrac{1}{2}.sin^22x\ge\dfrac{1}{2}\)\(\Leftrightarrow1\ge y\ge\dfrac{1}{2}\)
\(y_{min}=\dfrac{1}{2}\Leftrightarrow sin^22x=1\)\(\Leftrightarrow\)\(\left[{}\begin{matrix}sin2x=-1\\sin2x=1\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{4}+k\pi\end{matrix}\right.\) \(\left(k\in Z\right)\)
\(y_{max}=1\Leftrightarrow sin2x=0\Leftrightarrow x=\dfrac{k\pi}{2}\)\(\left(k\in Z\right)\)
c)\(y=\left(sin^2x+cos^2x\right)^3-3sin^2x.cos^2x\left(sin^2x+cos^2x\right)=1-3sin^2x.cos^2x=1-\dfrac{3}{4}.sin^22x\)
Có \(0\le sin^22x\le1\)\(\Leftrightarrow0\ge-\dfrac{3}{4}.sin^22x\ge-\dfrac{3}{4}\)
\(\Leftrightarrow1\ge1-\dfrac{3}{4}.sin^22x\ge\dfrac{1}{4}\)\(\Leftrightarrow1\ge y\ge\dfrac{1}{4}\)
\(y_{min}=\dfrac{1}{4}\Leftrightarrow sin^22x=1\)\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+k\pi\\x=-\dfrac{\pi}{4}+k\pi\end{matrix}\right.\)\(\left(k\in Z\right)\)
\(y_{max}=1\Leftrightarrow sin2x=0\Leftrightarrow x=\dfrac{k\pi}{2}\)\(\left(k\in Z\right)\)
Vậy...
a, Đặt \(t=sinx\left(t\in\left[-1;1\right]\right)\)
\(y=f\left(t\right)=\left(3-t\right)^2+1=t^2-6t+10\)
\(\Rightarrow min=min\left\{f\left(-1\right);f\left(1\right)\right\}=f\left(1\right)=5\)
\(\Rightarrow max=max\left\{f\left(-1\right);f\left(1\right)\right\}=f\left(-1\right)=17\)
b, \(y=sin^4x+cos^4x=1-2sin^2x.cos^2x=1-\dfrac{1}{2}sin^22x\)
Đặt \(t=sin2x\left(t\in\left[-1;1\right]\right)\)
\(y=f\left(t\right)=1-\dfrac{1}{2}t^2\)
\(\Rightarrow min=min\left\{f\left(-1\right);f\left(0\right);f\left(1\right)\right\}=\dfrac{1}{2}\)
\(\Rightarrow max=max\left\{f\left(-1\right);f\left(0\right);f\left(1\right)\right\}=1\)
c, \(y=sin^6x+cos^6x\)
\(=sin^4x+cos^4x-sin^2x.cos^2x\)
\(=1-3sin^2x.cos^2x\)
\(=1-\dfrac{3}{4}sin^22x\)
Đặt \(t=sin2x\left(t\in\left[-1;1\right]\right)\)
\(y=f\left(t\right)=1-\dfrac{3}{4}t^2\)
\(\Rightarrow min=min\left\{f\left(-1\right);f\left(0\right);f\left(1\right)\right\}=\dfrac{1}{4}\)
\(\Rightarrow max=max\left\{f\left(-1\right);f\left(0\right);f\left(1\right)\right\}=1\)
Đặt \(sin^24x=t\left(t\in\left[0;1\right]\right)\)
\(y=1-8sin^22x.cos^22x+2sin^42x\)
\(=1-2sin^24x+2sin^42x\)
\(\Rightarrow y=f\left(t\right)=1-2t+2t^2\)
\(y_{min}=min\left\{f\left(0\right);f\left(1\right);f\left(\dfrac{1}{2}\right)\right\}=\dfrac{1}{2}\)
\(y_{max}=max\left\{f\left(0\right);f\left(1\right);f\left(\dfrac{1}{2}\right)\right\}=1\)
Đề là:
\(y=\sqrt{4-3cos^23x}+1\) đúng không nhỉ?
Ta có:
\(0\le cos^23x\le1\Rightarrow1\le\sqrt{4-3cos^23x}\le2\)
\(\Rightarrow2\le y\le3\)
\(y_{min}=2\) khi \(cos^23x=1\)
\(y_{max}=3\) khi \(cos3x=0\)