Let the maximum and minimum value of the expression 2 cos 2 ⁡ θ + cos ⁡ θ + 1 is M and m respectively, then the value of M m is, (where [.] is the greatest integer function)

Step-by-step solution

2 cos 2 ⁡ θ + cos ⁡ θ + 1 = 2 cos 2 ⁡ θ + cos ⁡ θ 2 + 1 2 2 cos ⁡ θ + 1 4 2 + 7 16 Given expression is maximum when cos ⁡ θ = 1 and minimum when cos ⁡ θ = - 1 4 ⇒ M = 2 5 4 2 + 7 16 = 2 32 16 = 4 and m = 2 7 16 = 7 8 Hence, M m = 32 7 = 4