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a.\(-1\le cosx\le1\Rightarrow-4\le y=3cosx-1\le2\)
b.-1 \(\le sinx\le1\)\(\Rightarrow3\le y=5+2sinx\le7\)
c.\(\sqrt{3-1}\le\sqrt{3+cos2x}\le\sqrt{3+1}\Rightarrow\sqrt{2}\le y\le2\)
d.\(y=\sqrt{5sinx-1}+2\le\sqrt{5.1-1}+2=4\)
\(y=\sqrt{5sinx-1}+2\ge2\) . " = " \(\Leftrightarrow sinx=\dfrac{1}{5}\Leftrightarrow\left[{}\begin{matrix}x=arcsin\left(\dfrac{1}{5}\right)+2k\pi\\x=\pi-arcsin\left(\dfrac{1}{5}\right)+2k\pi\end{matrix}\right.\) ( k thuộc Z )
y=(sin2x-3)^2-6
-1<=sin2x<=1
=>-4<=sin2x-3<=-2
=>4<=(sin2x-3)^2<=16
=>-2<=y<=10
y min khi sin2x-3=-2
=>sin 2x=1
=>2x=pi/2+k2pi
=>x=pi/4+kpi
y max khi sin 2x-3=-4
=>sin 2x=-1
=>2x=-pi/2+k2pi
=>x=-pi/4+kpi
1.
\(y=\sqrt{5-2\cos ^2x\sin ^2x}=\sqrt{5-\frac{1}{2}(2\cos x\sin x)^2}=\sqrt{5-\frac{1}{2}\sin ^22x}\)
Dễ thấy:
$\sin ^22x\geq 0\Rightarrow y=\sqrt{5-\frac{1}{2}\sin ^22x}\leq \sqrt{5}$
Vậy $y_{\max}=\sqrt{5}$
$\sin ^22x\leq 1\Rightarrow y=\sqrt{5-\frac{1}{2}\sin ^22x}\geq \sqrt{5-\frac{1}{2}}=\frac{3\sqrt{2}}{2}$
Vậy $y_{\min}=\frac{3\sqrt{2}}{2}$
2.
$y=1+\frac{1}{2}\sin 2x\cos 2x=1+\frac{1}{4}.2\sin 2x\cos 2x$
$=1+\frac{1}{4}\sin 4x$
Vì $-1\leq \sin 4x\leq 1$
$\Rightarrow \frac{5}{4}\leq 1+\frac{1}{4}\sin 4x\leq \frac{3}{4}$
$\Leftrightarrow \frac{5}{4}\leq y\leq \frac{3}{4}$
Vậy $y_{\max}=\frac{5}{4}; y_{\min}=\frac{3}{4}$
a, Đồ thị hàm số \(y=cosx\): \(\left(A=\left(-\dfrac{\pi}{2};0\right);B=\left(\dfrac{\pi}{2};0\right)\right)\)
Dựa vào đồ thị ta có \(\left\{{}\begin{matrix}y_{min}=0\\y_{max}=1\end{matrix}\right.\)
b, Đồ thị hàm số \(y=sinx\): \(\left(A=\left(-\dfrac{\pi}{2};-1\right);A=\left(\dfrac{\pi}{2};1\right)\right)\)
`TXĐ: R`
Ta có: `-1 <= sin(x+ \pi/3) <= 1`
`<=>0 <= sin^4 (x+\pi/3) <= 1`
`<=>2 <= y <= 3`
`=>y_[mi n]=2<=>sin(x +\pi/3)=0<=>x= -\pi/3+k\pi` `(k in ZZ)`
`y_[max]=3<=>sin(x +\pi/3)=1<=>x=\pi/6 +k2\pi` `(k in ZZ)`
2.
$y=\sin ^4x+\cos ^4x=(\sin ^2x+\cos ^2x)^2-2\sin ^2x\cos ^2x$
$=1-\frac{1}{2}(2\sin x\cos x)^2=1-\frac{1}{2}\sin ^22x$
Vì: $0\leq \sin ^22x\leq 1$
$\Rightarrow 1\geq 1-\frac{1}{2}\sin ^22x\geq \frac{1}{2}$
Vậy $y_{\max}=1; y_{\min}=\frac{1}{2}$
3.
$0\leq |\sin x|\leq 1$
$\Rightarrow 3\geq 3-2|\sin x|\geq 1$
Vậy $y_{\min}=1; y_{\max}=3$
24.
\(cos\left(x-\dfrac{\pi}{2}\right)\le1\Rightarrow y\le3.1+1=4\)
\(y_{max}=4\)
26.
\(y=\sqrt{2}cos\left(2x-\dfrac{\pi}{4}\right)\)
Do \(cos\left(2x-\dfrac{\pi}{4}\right)\le1\Rightarrow y\le\sqrt{2}\)
\(y_{max}=\sqrt{2}\)
b.
\(\dfrac{1}{2}sinx+\dfrac{\sqrt{3}}{2}cosx=\dfrac{1}{2}\)
\(\Leftrightarrow cos\left(x-\dfrac{\pi}{6}\right)=\dfrac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{6}=\dfrac{\pi}{3}+k2\pi\\x-\dfrac{\pi}{6}=-\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k2\pi\\x=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)
Tìm TXĐ các hàm số:
a, y = sin \(2-\sqrt{x-1}\)
b, y = \(\dfrac{tanx}{cos2x+1}\)
c, y = \(\sqrt{cosx}\)
ĐKXĐ:
a. \(x-1\ge0\Rightarrow x\ge1\)
b. \(\left\{{}\begin{matrix}cosx\ne0\\cos2x+1\ne0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}cosx\ne0\\cos2x\ne-1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ne\dfrac{\pi}{2}+k\pi\\2x\ne\pi+k2\pi\end{matrix}\right.\) \(\Leftrightarrow x\ne\dfrac{\pi}{2}+k\pi\)
c.
\(cosx\ge0\Rightarrow-\dfrac{\pi}{2}+k2\pi\le x\le\dfrac{\pi}{2}+k2\pi\)
a/
Do \(-1\le sinx\le1\Rightarrow1\le3-2sinx\le5\)
\(y_{max}=5\) khi \(sinx=-1\)
\(y_{min}=1\) khi \(sinx=1\)
b.
Do \(0\le sin^22x\le1\Rightarrow3\le3+\dfrac{1}{8}sin^22x\le\dfrac{25}{8}\)
\(y_{min}=3\) khi \(sin2x=0\)
\(y_{max}=\dfrac{25}{8}\) khi \(sin^22x=1\)