If acosx+bcos3x≤1  ∀x∈R,  then  b

Given acosx+bcos3x≤1For x=0,           a+b≤1→i for          x=π3,     a2−b≤1→iifor          x=2π3,  -a2+b≤1→iiiand for         x=π,  −a−b≤1→ivFrom (i) and (iii) b≤1 and from (ii) and (iv) −b≤1⇒-1≤b≤1⇒b≤1