Задача 30618 ...

7gev

2018-10-29 14:46:57

Решить неравенство 20log^2_(4)(cosx)+4log_(2)(cosx) ≤ 1

sova

2018-10-29 14:57:16

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ОДЗ:
cosx > 0 ⇒ x ∈ ((-π/2)+2πn; (π/2)+2πn), n ∈ Z ( 1 и 4 четверти)

log_(4)cosx=log_(2^2)(cosx)=(1/2)log_(2)cosx
log^(2)_(4)cosx=(1/4)log^(2)_(2) cosx

5(log_(2)cosx)^2+4log_(2)cosx - 1 ≤ 0
D=16-4*5*(-1)=36
корни - 1 и 1/5
(5log_(2)cosx -1)*(log_(2)cosx+1) ≤ 0
-1 ≤ log_(2) cosx ≤ 1/5
log_(2)(1/2) ≤ log_(2) cosx ≤ log_(2) 2^(1/5) ⇒
(1/2) ≤ cosx≤ 2^(1/5)
⇒
(1/2) ≤ cosx
(-π/3)+2πm ≤ x ≤ (π/3)+2πm, m ∈ Z
о т в е т.
(-π/3)+2πm ≤ x ≤ (π/3)+2πm, m ∈ Z