Toán Lớp 11: sin^4(x/8)+cos^4(x/8)=2m-1 tìm m để pt có nghiệm trong khoảng (0;pi)

Question

Toán Lớp 11:  sin^4(x/8)+cos^4(x/8)=2m-1 tìm m để pt có nghiệm trong khoảng (0;pi)

truongthanhhung · Answer

$ begin{array}{l} x  in  left( {0; pi }  right)  Rightarrow  dfrac{x}{4}  in  left( {0; dfrac{ pi }{4}}  right)     Rightarrow 0  le  sin  left( { dfrac{x}{4}}  right)  le  dfrac{{ sqrt 2 }}{2}  Rightarrow 0  le { sin ^2} left( { dfrac{x}{4}}  right)  le  dfrac{1}{2}    left[ {{{ sin }^4} left( { dfrac{x}{8}}  right) + {{ cos }^4} left( { dfrac{x}{8}}  right)}  right] = 2m - 1     Leftrightarrow { left[ {{{ sin }^2} left( { dfrac{x}{8}}  right) + {{ cos }^2} left( { dfrac{x}{8}}  right)}  right]^2} - 2{ sin ^2} left( { dfrac{x}{8}}  right){ cos ^2} left( { dfrac{x}{8}}  right) = 2m - 1     Leftrightarrow 1 -  dfrac{1}{2}{ sin ^2} left( { dfrac{x}{4}}  right) = 2m - 1     Leftrightarrow  dfrac{1}{2}{ sin ^2} left( { dfrac{x}{4}}  right) = 2 - 2m     Leftrightarrow { sin ^2} left( { dfrac{x}{4}}  right) = 4 - 4m    text{Phương tr&igrave;nh c&oacute; nghiệm}  Leftrightarrow 0  le 4 - 4m  le  dfrac{1}{2}     Leftrightarrow  dfrac{7}{8}  le m  le 1  end{array}$