1) cos²x - (8 -√3)/2* cosx -2√3 =0 ; x∈(0; 2π) max(x) --?
cos²x = (4 -√3/2) *cosx -2√3 =0;
t =cosx ; |t|≤1 ;
t² - (4 -√3/2) t -2√3 =0;
D= (4 -√3/2)² +8√3=16 -4√3 +(√3/2)²+8√3 =16 +4√3 +(√3/2)² =(4+√3/2)² ;
t₁ =((4 -√3/2) -(4 +√3/2))/2 = -√3/2;
t ₂= (4 -√3/2) + (4 +√3/2))/2 =4 >1 .
cosx = -√3/2 ;
x =(+/-) 5π/6 +2π*k , k∈Z
x =5π/6 и x =7π/6 ∈(0; 2π)
max (x) =7π/6 или 210°
================================
2) cos²2x -1 -cosx =√3/2 -sin²2x ; x∈[ 0; 2π ]; max(x) --? ;
(cos²2x+-sin²2x) -1 -cosx =√3/2 ;
1 -1 -cosx =√3/2 ;
cosx = - √3/2;
max (x) =7π/6 или 210°, x∈[0 ;2π] . см пред
3) 2cos²x -sinx*cosx+5sin²x =3 ; x∈(0; 2π) min(x) --?
2cos²x -sinx*cosx+5sin²x =3(cos²x +sin²x);
2sin²x - sinx*cosx -cos²x =0 ;
cosx≠0 (иначе получилось бы и sinx =0 но sin²x +cos²x =1)
2tq²x -tqx -1 = 0;
tqx = -1/2;
tqx=1 ;
x = -arctq(1/2) +π*k ;
x =π/4 +π*k ;
x = π - arctq1/2 ,2π - arctq1/2 ; π/4 +π =5π/4 ∈ [0;2π]
min(x) = 5π/4 или 225.