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8.
\(y=\left(cosx+1\right)^2-1\ge-1\Rightarrow y_{min}=-1\)
\(y=\left(cosx-1\right)\left(cosx+3\right)+3\le3\Rightarrow y_{max}=3\)
10.
\(y=2-\left(cosx+1\right)^2\le2\Rightarrow y_{max}=2\)
14.
Hàm tuần hoàn với chu kì \(T=\pi\)
15.
Đáp án A đúng
20.
\(-1\le sin\left(\frac{x}{2}+\frac{\pi}{7}\right)\le1\Rightarrow-5\le y\le-1\)
\(y_{max}=-1\) ; \(y_{min}=-5\)
8.
\(y=cos^2x+2\left(2cos^2x-1\right)=5cos^2x-2\)
Do \(0\le cos^2x\le1\Rightarrow-2\le y\le3\)
\(y_{min}=-2;y_{max}=3\)
10.
\(y=2-\left(cosx+1\right)^2\le2\)
\(y_{max}=2\)
14.
Hàm tuần hoàn với chu kì \(T=\pi\)
Bài 1:
1: \(y=\frac{\sin x+2\cdot cosx+1}{2\cdot\sin x+cosx+3}\)
=>\(2y\cdot\sin x+y\cdot cosx+3y=\sin x+2\cdot cosx+1\)
=>\(\left(2y-1\right)\cdot\sin x+cosx\cdot\left(y-2\right)=1-3y\)
Để phương trình có nghiệm thì \(\left(2y-1\right)^2+\left(y-2\right)^2>=\left(1-3y\right)^2\)
=>\(4y^2-4y+1+y^2-4y+4\ge9y^2-6y+1\)
=>\(5y^2-8y+5-9y^2+6y-1\ge0\)
=>\(-4y^2-2y+4\ge0\)
=>\(y^2+\frac12y-1\le0\)
=>\(y^2+2\cdot y\cdot\frac14+\frac{1}{16}-\frac{17}{16}\le0\)
=>\(\left(y+\frac14\right)^2\le\frac{17}{16}\)
=>\(-\frac{\sqrt{17}}{4}\le y+\frac14\le\frac{\sqrt{17}}{4}\)
=>\(\frac{-\sqrt{17}-1}{4}\le y\le\frac{\sqrt{17}-1}{4}\)
=>\(y_{\min}=\frac{-\sqrt{17}-1}{4}\) và \(y_{\max}=\frac{\sqrt{17}-1}{4}\)
2: \(y=2\cdot\sin^2x-3\cdot\sin x\cdot cosx+cos^2x\)
\(=2\cdot\frac{1-cos2x}{2}-3\cdot\frac12\cdot\sin2x+\frac{1+cos2x}{2}\)
\(=1-cos2x-\frac32\cdot\sin2x+\frac12+\frac12\cdot cos2x\)
\(=-\frac32\cdot\sin2x-\frac12\cdot cos2x+\frac32=-\frac12\left(3\cdot\sin2x+cos2x-3\right)\)
\(=-\frac{\sqrt{10}}{2}\left(\frac{3}{\sqrt{10}}\cdot\sin2x+\frac{1}{\sqrt{10}}\cdot cos2x-\frac{3}{\sqrt{10}}\right)\)
\(=-\frac{\sqrt{10}}{2}\cdot\left\lbrack\sin\left(2x+\alpha\right)-\frac{3}{\sqrt{10}}\right\rbrack\) , với \(cosa=\frac{3}{\sqrt{10}};\sin a=\frac{1}{\sqrt{10}}\)
\(=-\frac{\sqrt{10}}{2}\cdot\sin\left(2x+\alpha\right)+\frac32\)
Ta có: \(-1\le\sin\left(2x+a\right)\le1\)
=>\(-1\cdot\frac{-\sqrt{10}}{2}\ge\frac{-\sqrt{10}}{2}\sin\left(2x+a\right)\ge1\cdot\frac{-\sqrt{10}}{2}\)
=>\(\frac{-\sqrt{10}}{2}\le\frac{-\sqrt{10}}{2}\cdot\sin\left(2x+a\right)\le\frac{\sqrt{10}}{2}\)
=>\(\frac{-\sqrt{10}}{2}+\frac32\le\frac{-\sqrt{10}}{2}\cdot\sin\left(2x+a\right)+\frac32\le\frac{\sqrt{10}}{2}+\frac32\)
=>\(y_{\min}=\frac{-\sqrt{10}+3}{2};y_{\max}=\frac{\sqrt{10}+3}{2}\)
1: \(-1<=cosx\le1\)
=>\(-3\le-3\cdot cosx\le3\)
=>\(-3+5\le-3\cdot cosx+5\le3+5\)
=>2<=y<=8
y min=2 khi cosx=1
=>\(x=k2\pi\)
y min=8 khi cosx=-1
=>\(x=\pi+k2\pi\)
3: \(y=cos^2x+2\cdot cos2x\)
\(=\frac{1+cos2x}{2}+2\cdot cos2x=2,5\cdot cos2x+0,5\)
Ta có: \(-1\le cos2x\le1\)
=>\(-2,5\le2,5cos2x\le2,5\)
=>\(-2,5+0,5\le2,5cos2x+0,5\le2,5+0,5\)
=>-2<=y<=3
y min=-2 khi cos2x=-1
=>\(2x=\pi+k2\pi\)
=>\(x=\frac{\pi}{2}+k\pi\)
y max=3 khi cos2x=1
=>\(2x=k2\pi\)
=>\(x=k\pi\)
6: \(y=\sqrt3\cdot\sin x-cosx-2\)
\(=2\left(\frac{\sqrt3}{2}\cdot\sin x-\frac12\cdot cosx\right)-2=2\cdot\sin\left(x-\frac{\pi}{6}\right)-2\)
Ta có: \(-1\le\sin\left(x-\frac{\pi}{6}\right)\le1\)
=>\(-2\le2\sin\left(x-\frac{\pi}{6}\right)\le2\)
=>\(-2-2\le2\sin\left(x-\frac{\pi}{6}\right)-2\le2-2\)
=>-4<=y<=0
y min=-4 khi \(\sin\left(x-\frac{\pi}{6}\right)=-1\)
=>\(x-\frac{\pi}{6}=-\frac{\pi}{2}+k2\pi\)
=>\(x=-\frac{\pi}{2}+\frac{\pi}{6}+k2\pi=-\frac26\pi+k2\pi=-\frac13\pi+k2\pi\)
y max=0 khi \(\sin\left(x-\frac{\pi}{6}\right)=1\)
=>\(x-\frac{\pi}{6}=\frac{\pi}{2}+k2\pi\)
=>\(x=\frac23\pi+k2\pi\)
4.
\(-1\le sinx\le1\Rightarrow\sqrt{2}\le\sqrt{sinx+3}\le2\)
\(\Rightarrow\sqrt{2}-1\le y\le1\)
12.
ĐKXĐ: \(sinx+1\ne0\Rightarrow sinx\ne-1\Rightarrow x\ne-\frac{\pi}{2}+k2\pi\)
20.
\(y=tanx\) ko xác định khi \(cosx=0\Leftrightarrow x=\frac{\pi}{2}+k\pi\)
22.
\(y_{min}=y\left(\frac{\pi}{4}\right)=cos\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\)
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 $y_{\min}=1; y_{\max}=3$
a/ \(y=sin2x+\left(\sqrt{3}+1\right)cos2x+sin^2x-cos^2x-1\)
\(=sin2x+\sqrt{3}cos2x-1=2sin\left(2x+\frac{\pi}{3}\right)-1\)
Do \(-1\le sin\left(2x+\frac{\pi}{3}\right)\le1\Rightarrow-3\le y\le1\)
b/ \(y=2sin^2x-2cos^2x-3sinx.cosx-1\)
\(=-2cos2x-\frac{3}{2}sin2x-1=-\frac{5}{2}\left(\frac{3}{5}sinx+\frac{4}{5}cosx\right)-1\)
\(=-\frac{5}{2}sin\left(x+a\right)-1\Rightarrow-\frac{7}{2}\le y\le\frac{3}{2}\)
c/ \(y=1-sin2x+2cos2x+\frac{3}{2}sin2x=\frac{1}{2}sin2x+2cos2x+1\)
\(=\frac{\sqrt{17}}{2}\left(\frac{1}{\sqrt{17}}sin2x+\frac{4}{\sqrt{17}}cos2x\right)+1=\frac{\sqrt{17}}{2}sin\left(2x+a\right)+1\)
\(\Rightarrow-\frac{\sqrt{17}}{2}+1\le y\le\frac{\sqrt{17}}{2}+1\)
2.
\(0\le\left|sinx\right|\le1\Rightarrow1\le y\le3\)
Min và max lần lượt là 3 và 1
3.
\(cos\left(x-\frac{\pi}{2}\right)\le1\Rightarrow y\le3.1+1=4\)
8.
\(y=\frac{1}{2}+\frac{1}{2}cos2x+2cos2x=\frac{1}{2}+\frac{5}{2}cos2x\le\frac{1}{2}+\frac{5}{2}.1=3\)
15.
Nó đi qua vô số điểm nên ko có 4 đáp án để chọn thì ko ai có thể trả lời câu này cho bạn cả
18.
\(y=\frac{sinx+2cosx+1}{sinx+cosx+2}\Leftrightarrow y.sinx+y.cosx+2y=sinx+2cosx+1\)
\(\Leftrightarrow\left(y-1\right)sinx+\left(y-2\right)cosx=1-2y\)
\(\left(y-1\right)^2+\left(y-2\right)^2\ge\left(1-2y\right)^2\)
\(\Leftrightarrow2y^2+2y-4\le0\Rightarrow-2\le y\le1\)
\(\Rightarrow y_{max}=1\)