Bài 3:
a: \(-1\le cosx\le1\)
=>-3<=3cosx<=3
=>-3+2<=3cosx+2<=3+2
=>-1<=y<=5
y min=-1 khi cosx=-1
=>\(x=\pi+k2\pi\)
y max=5 khi cosx=1
=>\(x=k2\pi\)
b: \(-1\le\sin\left(x-\frac{\pi}{6}\right)\le1\)
=>\(-3\le3\cdot\sin\left(x-\frac{\pi}{6}\right)\le3\)
=>\(-3-2\le3\cdot\sin\left(x-\frac{\pi}{6}\right)-2\le3-2\)
=>-5<=y<=1
y min=-5 khi \(\sin\left(x-\frac{\pi}{6}\right)=-1\)
=>\(x-\frac{\pi}{6}=-\frac{\pi}{2}+k2\pi\)
=>\(x=-\frac{\pi}{2}+\frac{\pi}{6}+k2\pi=-\frac{\pi}{3}+k2\pi\)
y max=1 khi \(\sin\left(x-\frac{\pi}{6}\right)=1\)
=>\(x-\frac{\pi}{6}=\frac{\pi}{2}+k2\pi\)
=>\(x=\frac{\pi}{2}+\frac{\pi}{6}+k2\pi=\frac46\pi+k2\pi=\frac23\pi+k2\pi\)
c: \(0\le cos^22x\le1\)
=>\(0\le4\cdot cos^22x\le4\)
=>\(1\le4\cdot cos^22x+1\le5\)
=>\(1\le\sqrt{4\cdot cos^22x+1}\le\sqrt5\)
=>\(1\le y\le\sqrt5\)
y min=1 khi \(cos^22x=0\)
=>cos2x=0
=>\(2x=\frac{\pi}{2}+k\pi\)
=>\(x=\frac{\pi}{4}+\frac{k\pi}{2}\)
y max=\(\sqrt5\) khi \(cos^22x=1\)
=>\(\sin^22x=0\)
=>sin 2x=0
=>\(2x=k\pi\)
=>\(x=\frac{k\pi}{2}\)
d: \(0\le\left|\sin x\right|\le1\)
=>\(0\ge-2\cdot\left|\sin x\right|\ge-2\)
=>0+3>=-2|sinx|+3>=-2+3
=>3>=y>=1
y max=3 khi |sin x|=0
=>sin x=0
=>\(x=k\pi\)
y min=1 khi |sin x|=1
=>\(\left[\begin{array}{l}\sin x=1\\ \sin x=-1\end{array}\right.\Rightarrow\left[\begin{array}{l}x=\frac{\pi}{2}+k2\pi\\ x=-\frac{\pi}{2}+k2\pi\end{array}\right.\)
f: \(y=\sin x+cosx\)
\(=\sqrt2\cdot\sin\left(x+\frac{\pi}{4}\right)\)
Ta có: \(-1\le\sin\left(x+\frac{\pi}{4}\right)\le1\)
=>\(-\sqrt2\le\sqrt2\cdot\sin\left(x+\frac{\pi}{4}\right)\le\sqrt2\)
=>\(-\sqrt2\le y\le\sqrt2\)
y min=-\(\sqrt2\) khi \(\sin\left(x+\frac{\pi}{4}\right)=-1\)
=>\(x+\frac{\pi}{4}=-\frac{\pi}{2}+k2\pi\)
=>\(x=-\frac{\pi}{2}-\frac{\pi}{4}+k2\pi=-\frac34\pi+k2\pi\)
y max=\(\sqrt2\) khi \(\sin\left(x+\frac{\pi}{4}\right)=1\)
=>\(x+\frac{\pi}{4}=\frac{\pi}{2}+k2\pi\)
=>\(x=\frac{\pi}{4}+k2\pi\)
Bài 2:
a: TXĐ là D=R
Khi x∈D thì -x∈D
\(f\left(-x\right)=2\cdot\left(-x\right)\cdot\sin\left(-x\right)=-2x\cdot\left\lbrack-\sin x\right\rbrack=2x\cdot\sin x=f\left(x\right)\)
=>f(x) là hàm số chẵn
b: TXĐ là D=R
Khi x∈D thì -x∈D
\(f\left(-x\right)=cos\left(-x\right)+\sin\left(-2x\right)=cosx-\sin2x\)
=>f(-x)<>-f(x) và f(-x)<>f(x)
=>f(x) không là hàm số chẵn và cũng không là hàm số lẻ
c: TXĐ là D=R\{0}
Khi x∈D thì -x∈D
\(f\left(-x\right)=\frac{cos\left(-2x\right)}{-x}=\frac{cos2x}{-x}=-f\left(x\right)\)
=>f(x) là hàm số lẻ
d: ĐKXĐ: \(2x<>\frac{\pi}{2}+k\pi\)
=>\(x<>\frac{\pi}{4}+\frac{k\pi}{2}\)
=>TXĐ là D=R\{\(\frac{\pi}{4}+\frac{k\pi}{2}\) }
Khi x∈D thì -x∈D
\(f\left(-x\right)=\tan^7\left(-2x\right)\cdot\sin\left(-5x\right)=\tan^72x\cdot\sin\left(5x\right)=f\left(x\right)\)
=>f(x) là hàm số chẵn
