Let the function f : 0 , π → ℝ be defined by f θ = sin θ + cos θ 2 + sin θ - cos θ 4 . Suppose the function f has a local minimum at θ precisely when θ ∈ λ 1 π , … , λ r π , where 0

Question

Let the function f : 0 , π → ℝ be defined by f θ = sin θ + cos θ 2 + sin θ - cos θ 4 . Suppose the function f has a local minimum at θ precisely when θ ∈ λ 1 π , … , λ r π , where 0 < λ 1 < ⋯ < λ r < 1 . Then the value of λ 1 + ⋯ + λ r is_______

TestHub · Accepted Answer

f θ = 1 + sin 2 θ + 1 - sin 2 θ 2 = sin 2 2 θ - sin 2 θ + 2 = sin 2 θ - 1 2 2 + 7 4 It has minima when sin 2 θ = 1 2 = sin π 6 θ ∈ 0 , π 2 θ = π 6 , 5 π 6 θ = π 12 , 5 π 12 λ 1 = 1 12 , λ 2 = 5 12 sum = 1 2 .

Mathematics - Application of Derivative Question with Solution | TestHub

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