Question

Phương pháp đánh giá

Answer 1

Áp dụng BĐT bunhiacopxki có:

\(\left(sin^42x+cos^42x\right)\left(1+1\right)\ge\left(sin^22x+cos^22x\right)^2\)\(\Rightarrow sin^42x+cos^42x\ge\dfrac{1}{2}\)

\(\left(sin^82x+cos^82x\right)\left(1+1\right)\ge\left(sin^42x+cos^42x\right)^2\ge\dfrac{1}{4}\)\(\Rightarrow sin^82x+cos^82x\ge\dfrac{1}{8}\)

Dấu "=" xảy ra khi \(sin^22x=cos^22x=\dfrac{1}{2}\)\(\Rightarrow sin^22x.cos^22x=\dfrac{1}{4}\)

\(\Leftrightarrow sin^24x=1\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{8}+\dfrac{k\pi}{2}\\x=-\dfrac{\pi}{8}+\dfrac{k\pi}{2}\end{matrix}\right.\)

Vậy...