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a: -1<=sin x<=1
=>-1+3<=sin x+3<=1+3
=>2<=sinx+3<=4
=>\(\dfrac{1}{2}>=\dfrac{1}{sinx+3}>=\dfrac{1}{4}\)
=>\(2>=\dfrac{4}{sinx+3}>=1\)
=>\(-2< =-\dfrac{4}{sinx+3}< =-1\)
=>-2+3<=y<=-1+3
=>1<=y<=2
y=1 khi \(\dfrac{-4}{sinx+3}+3=1\)
=>\(\dfrac{-4}{sinx+3}=-2\)
=>sinx+3=2
=>sin x=-1
=>x=-pi/2+k2pi
y=3 khi sin x=1
=>x=pi/2+k2pi
b: -1<=cosx<=1
=>4>=-4cosx>=-4
=>9>=-4cosx+5>=1
=>2/9<=2/5-4cosx<=2
=>2/9<=y<=2
\(y_{min}=\dfrac{2}{9}\) khi \(\dfrac{2}{5-4cosx}=\dfrac{2}{9}\)
=>\(5-4\cdot cosx=9\)
=>4*cosx=4
=>cosx=1
=>x=k2pi
y max khi cosx=-1
=>x=pi+k2pi
c: \(0< =cos^2x< =1\)
=>\(0< =2\cdot cos^2x< =2\)
=>\(-1< =y< =2\)
y min=-1 khi cos^2x=0
=>x=pi/2+kpi
y max=2 khi cos^2x=1
=>sin^2x=0
=>x=kpi
b: \(-1< =cos4x< =1\)
=>\(-3< =3\cdot cos4x< =3\)
=>\(9< =3\cdot cos4x+12< =15\)
=>\(3< =y< =\sqrt{15}\)
y min=3 khi cos4x=-1
=>4x=pi+k2pi
=>x=pi/4+kpi/2
y max=căn 15 khi cos4x=1
=>4x=k2pi
=>x=kpi/2
c: -1<=sin 9x<=1
=>-1+20<=sin 9x+20<=21
=>19<=y<=21
y min=19 khi sin 9x=-1
=>9x=-pi/2+k2pi
=>x=-pi/18+k2pi/9
y max=21 khi sin 9x=1
=>9x=pi/2+k2pi
=>x=pi/18+k2pi/9
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 \(tanx=1\Rightarrow x=\dfrac{\pi}{4}+k\pi\)
\(y=\sqrt{\left(sinx+cosx\right)^2+2\cdot sinx\cdot cosx+2}\)
\(=\sqrt{1+2sinx\cdot cosx+2\cdot sinx\cdot cosx+2}\)
\(=\sqrt{3+2sin2x}\)
\(-1< =sin2x< =1\)
=>\(-2< =2\cdot sin2x< =2\)
=>\(-2+3< =2\cdot sin2x+3< =5\)
=>\(1< =2\cdot sin2x+3< =5\)
=>\(1< =\sqrt{2\cdot sin2x+3}< =\sqrt{5}\)
=>\(1< =y< =\sqrt{5}\)
\(y_{min}=1\) khi \(sin2x=-1\)
=>\(2x=-\dfrac{\Omega}{2}+k2\Omega\)
=>\(x=-\dfrac{\Omega}{4}+k\Omega\)
\(y_{max}=\sqrt{5}\) khi sin 2x=1
=>\(2x=\dfrac{\Omega}{2}+k2\Omega\)
=>\(x=\dfrac{\Omega}{4}+k\Omega\)
a: \(0< =cos^23x< =1\)
=>\(9< =cos^23x+9< =10\)
=>9<=y<=10
\(y_{min}=9\) khi \(cos^23x=0\)
=>\(cos3x=0\)
=>3x=pi/2+kpi
=>x=pi/6+kpi/3
\(y_{max}=10\) khi \(cos^23x=0\)
=>\(sin^23x=0\)
=>3x=kpi
=>x=kpi/3
b: \(0< =sin^2x< =1\)
=>\(-3< =y< =-2\)
\(y_{min}=-3\) khi \(sin^2x=0\)
=>x=kpi
\(y_{max}=-2\) khi \(sin^2x=1\)
=>\(cos^2x=0\)
=>x=pi/2+kpi
c: \(0< =sin^25x< =1\)
=>12<=y<=13
y min=12 khi sin25x=0
=>sin 5x=0
=>5x=kpi
=>x=kpi/5
y max=13 khi sin25x=0
=>cos25x=0
=>cos5x=0
=>5x=pi/2+kpi
=>x=pi/10+kpi/5
a: -1<=sinx<=1
=>5>=-5sinx>=-5
=>11>=-5sinx+6>=1
=>1<=y<=11
\(y_{min}=1\) khi sin x=1
=>x=pi/2+k2pi
\(y_{max}=11\) khi sin x=-1
=>x=-pi/2+k2pi
b: \(-1< =cosx< =1\)
=>\(1>=-cosx>=-1\)
=>\(-3>=-cosx-4>=-5\)
=>\(-3>=y>=-5\)
\(y_{min}=-5\) khi cosx=1
=>x=k2pi
\(y_{max}=-3\) khi cosx=-1
=>x=pi+k2pi
c: \(-1< =cosx< =1\)
=>\(-\sqrt{3}< \sqrt{3}\cdot cosx< =\sqrt{3}\)
=>\(-\sqrt{3}+8< =y< =\sqrt{3}+8\)
\(y_{min}=-\sqrt{3}+8\) khi cosx=-1
=>x=pi+k2pi
\(y_{max}=\sqrt{3}+8\) khi cosx=1
=>x=k2pi
d: \(-1< =cos3x< =1\)
=>\(1>=-cos3x>=-1\)
=>\(16>=y>=14\)
y min=14 khi cos3x=1
=>3x=k2pi
=>x=k2pi/3
y max=16 khi cos3x=-1
=>3x=pi+k2pi
=>x=pi/3+k2pi/3
e: -1<=sin6x<=1
=>-1+2024<=sin6x+2024<=1+2024
=>2023<=y<=2025
y min=2023 khi sin6x=-1
=>6x=-pi/2+k2pi
=>x=-pi/12+kpi/3
y max=2025 khi sin6x=1
=>6x=pi/2+k2pi
=>x=pi/12+kpi/3