a: \(y=3\cdot sinx+4\cdot cosx+1000\)
\(=5\left(\dfrac{3}{5}\cdot sinx+\dfrac{4}{5}\cdot cosx\right)+1000\)
\(=5\cdot\left(sinx\cdot cosa+cosx\cdot sina\right)+1000\)(Vì \(\left(\dfrac{3}{5}\right)^2+\left(\dfrac{4}{5}\right)^2=1\) nên \(sina=\dfrac{3}{5};cosa=\dfrac{4}{5}\))
\(=5\cdot sin\left(x+a\right)+1000\)
\(-1< =sin\left(x+a\right)< =1\)
=>\(-5< =5\cdot sin\left(x+a\right)< =5\)
=>\(-5+1000< =5\cdot sin\left(x+a\right)+1000< =1005\)
=>\(995< =y< =1005\)
Vậy: TGT là T=[995;1005]
b: TH1: sin x=0
=>\(x=k\Omega\)
Khi \(x=k\Omega\) thì \(cosx\cdot cos2x\cdot cos4x\cdot cos8x\)
\(=cos\left(k\Omega\right)\cdot cos\left(2\cdot k\Omega\right)\cdot cos\left(4\cdot k\Omega\right)\cdot cos\left(8\cdot k\Omega\right)\)
\(=\pm1\)
=>Trường hợp này loại
TH2: sin x<>0
\(cosx\cdot cos2x\cdot cos4x\cdot cos8x=\dfrac{1}{16}\)
=>\(2\cdot sinx\cdot cosx\cdot cos2x\cdot cos4x\cdot cos8x=\dfrac{1}{16}\cdot2\cdot sinx\)
=>\(sin2x\cdot cos2x\cdot cos4x\cdot cos8x=\dfrac{1}{8}\cdot sinx\)
=>\(2\cdot sin2x\cdot cos2x\cdot cos4x\cdot cos8x=\dfrac{1}{8}\cdot2\cdot sinx\)
=>\(sin4x\cdot cos4x\cdot cos8x=\dfrac{1}{4}\cdot sinx\)
=>\(2\cdot sin4x\cdot cos4x\cdot cos8x=\dfrac{1}{4}\cdot2\cdot sinx\)
=>\(sin8x\cdot cos8x=\dfrac{1}{2}\cdot sinx\)
=>\(2\cdot sin8x\cdot cos8x=sinx\)
=>\(sin16x=sinx\)
=>\(\left[{}\begin{matrix}16x=x+k2\Omega\\16x=\Omega-x+k2\Omega\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=\dfrac{k2\Omega}{15}\\x=\dfrac{\Omega}{17}+\dfrac{k2\Omega}{17}\end{matrix}\right.\)
