\(y=\dfrac{4}{3}\left(\sin^6x+\cos^6x\right)+\cos4x-1\)

\(\sin^6x+\cos^6x=\left(\sin^2x+\cos^2x\right)\left(\sin^4x-\sin^2x\cdot\cos^2x+\cos^4x\right)\\ =\left(\sin^2x+\cos^2x\right)^2-3\sin^2x\cdot\cos^2x=1-\dfrac{3}{4}\sin^22x\)

Do \(0\le\sin^22x\le1\Leftrightarrow\dfrac{3}{4}\cdot0\ge-\dfrac{3}{4}\sin^22x\ge-\dfrac{3}{4}\)

\(\Leftrightarrow1\ge1-\dfrac{3}{4}\sin^22x\ge1-\dfrac{3}{4}=\dfrac{1}{4}\\ \Leftrightarrow\dfrac{4}{3}\ge\dfrac{4}{3}\left(\sin^6x+\cos^6x\right)\ge\dfrac{1}{4}\cdot\dfrac{4}{3}=\dfrac{1}{3}\)

Ta có \(-1\le\cos4x\le1\)

\(\Leftrightarrow\dfrac{1}{3}-1-1\le\dfrac{4}{3}\left(\sin^6x+\cos^6x\right)+\cos4x-1\le\dfrac{4}{3}+1-1\\ \Leftrightarrow-\dfrac{5}{3}\le y\le\dfrac{4}{3}\)

Vậy \(y_{min}=-\dfrac{5}{3};y_{max}=\dfrac{4}{3}\)

\(y=\dfrac{4}{3}\left(sin^6x+cos^6x\right)+cos4x-1\)

\(y=\dfrac{4}{3}\left(\dfrac{5}{8}+\dfrac{3}{8}cos4x\right)+cos4x-1\)

\(y=\dfrac{3}{2}cos4x-\dfrac{1}{6}\)

\(-1\le cos4x\le1\Rightarrow-\dfrac{5}{3}\le y\le\dfrac{4}{3}\)

\(y_{min}=-\dfrac{5}{3}\) khi \(cos4x=-1\)

\(y_{max}=\dfrac{4}{3}\) khi \(cos4x=1\)

b: \(y=\sin x+\sqrt3\cdot cosx\)

\(=2\left(\frac12\cdot\sin x+\frac{\sqrt3}{2}\cdot cosx\right)\)

\(=2\cdot\sin\left(x+\frac{\pi}{3}\right)\)

TA có: \(-1\le\sin\left(x+\frac{\pi}{3}\right)\le1\)

=>\(-2\le2\cdot\sin\left(x+\frac{\pi}{3}\right)\le2\)

=>-2<=y<=2

y min=-2 khi \(\sin\left(x+\frac{\pi}{3}\right)=-1\)

=>\(x+\frac{\pi}{3}=-\frac{\pi}{2}+k2\pi\)

=>\(x=-\frac56\pi+k2\pi\)

y max=2 khi \(\sin\left(x+\frac{\pi}{3}\right)=1\)

=>\(x+\frac{\pi}{3}=\frac{\pi}{2}+k2\pi\)

=>\(x=\frac{\pi}{2}-\frac{\pi}{3}+k2\pi=\frac{\pi}{6}+k2\pi\)

c: 0<=|cosx|<=1

=>0>=-2|cosx|>=-2

=>0+3>=-2|cosx|+3>=-2+3

=>3>=y>=1

y max=3 khi cosx=0

=>\(x=\frac{\pi}{2}+k\pi\)

y min=1 khi |cosx|=1

=>\(cos^2x=1\)

=>\(\sin^2x=0\)

=>sin x=0

=>\(x=k\pi\)

Chọn A.

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`

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