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 \(T=\pi\)

1.

Các hàm \(sinx;sin\frac{x}{2};sin\frac{x}{3};...;sin\frac{x}{10}\) có chu kì lần lượt là \(2\pi;4\pi;6\pi;...;20\pi\)

\(\Rightarrow\) Chu kì của hàm đã cho là \(BCNN\left(2\pi;4\pi;...;20\pi\right)=15120\pi\)

2.

a.

\(y=cos^22x+3cos2x+3\)

\(y=\left(cos2x+1\right)\left(cos2x+2\right)+1\ge1\Rightarrow y_{min}=1\) khi \(cos2x=-1\)

\(y=\left(cos2x-1\right)\left(cos2x+4\right)+7\le7\Rightarrow y_{max}=7\) khi \(cos2x=1\)

b.

Đặt \(a=4sinx-3cosx\Rightarrow a^2\le\left(4^2+\left(-3\right)^2\right)\left(sin^2x+cos^2x\right)=25\)

\(\Rightarrow-5\le a\le5\)

\(y=a^2-4a+1\) với \(a\in\left[-5;5\right]\)

\(y=\left(a-2\right)^2-3\ge-3\Rightarrow y_{min}=-3\) khi \(a=2\)

\(y=\left(a-9\right)\left(a+5\right)+46\le46\Rightarrow y_{max}=46\) khi \(a=-5\)

Em ko hiểu câu 2a

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 ) 

\(y=\sqrt{3}cosx-sinx=2\left(\dfrac{\sqrt{3}}{2}cosx-\dfrac{1}{2}sinx\right)=2cos\left(x+\dfrac{\pi}{6}\right)\)

Vì \(cos\left(x+\dfrac{\pi}{6}\right)\in\left[-1;1\right]\Rightarrow y=\sqrt{3}cosx-sinx\in\left[-2;2\right]\)

\(\Rightarrow y_{min}=-2\Leftrightarrow cos\left(x+\dfrac{\pi}{6}\right)=-1\Leftrightarrow x+\dfrac{\pi}{6}=\pi+k2\pi\Leftrightarrow x=\dfrac{5\pi}{6}+k2\pi\)

\(y_{max}=2\Leftrightarrow cos\left(x+\dfrac{\pi}{6}\right)=1\Leftrightarrow x+\dfrac{\pi}{6}=k2\pi\Leftrightarrow x=-\dfrac{\pi}{6}+k2\pi\)

a) Ta có : -\(\sqrt{a^2+b^2}< =asinx+bcosx< =\sqrt{a^2+b^2}\)

=> \(-\sqrt{12^2+\left(-5\right)^2}< =y< =\sqrt{12^2+\left(-5\right)^2}\)

<=> \(-\sqrt{13}< =y< =\sqrt{13}\)

Vậy min=\(-\sqrt{13}\) ,max=\(\sqrt{13}\)

b) \(-\sqrt{9+16}< =3cosx-4sinx< =\sqrt{9+16}\)

<=> -5 <=3cos x -4sinx <= 5

<=> 0<= y <= 10

Vậy min=0 max=10

`y=1/2 sinx +3cosx`

`-\sqrt( (1/2)^2+3^2) <= y <= \sqrt( (1/2)^2+3^2)`

`<=> -\sqrt37/2 <= y <= \sqrt37/2`

`=> y_(min) = -\sqrt37/2`

`y_(max) = \sqrt37/2`.

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 \(\sin\left(x-\frac{\pi}{6}\right)=1\)

=>\(x-\frac{\pi}{6}=\frac{\pi}{2}+k2\pi\)

=>\(x=\frac23\pi+k2\pi\)