Let the function f:0,π→ℝ be defined by fθ=sinθ+cosθ^2+sinθ-cosθ^4. Suppose the function f has a local minimum at θ precisely when θ∈λ_1π,…,λ_rπ, where…

Question

Let the function f:0,π→ℝ be defined by fθ=sinθ+cosθ^2+sinθ-cosθ^4. Suppose the function f has a local minimum at θ precisely when θ∈λ_1π,…,λ_rπ, where 0<λ_1<⋯<λ_r<1. Then the value of λ_1+⋯+λ_r is_______

MARKS Content Team · Accepted Answer

Correct answer: 0.5. fθ=1+sin2θ+1-sin2θ^2 =sin^22θ-sin2θ+2 =sin2θ-12^2+7/4 It has minima when sin2θ=1/2=sin(π)/6 θ∈0,π 2θ=(π)/6,(5π)/6 θ=(π)/12,(5π)/12 λ_1=1/12, λ_2=5/12 sum=1/2.

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Let the function f:0,π→ℝ be defined by fθ=sinθ+cosθ2+sinθ-cosθ4. Suppose the function f has a local minimum at θ precisely when θ∈λ1π,…,λrπ, where 0<λ1<⋯<λr<1. Then the value of λ1+⋯+λr is_______

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