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∈λ1π,…,λrπ where 0<λ1<⋯<λr<1 . Then the value of λ1+⋯+λr is

f(θ)=(sin⁡θ+cos⁡θ)2+(sin⁡θ−cos⁡θ)4  = 1+sin2θ+ 1-sin2θ2f(θ)=sin2⁡2θ−sin⁡2θ+2f′(θ)=2(sin⁡2θ)⋅(2cos⁡2θ)−2cos⁡2θ =2cos⁡2θ(2sin⁡2θ−1)f'θ=0 ⇒sin 2θ=12 or cos 2θ=0                          2θ=π6, 5π6  or        2θ=π2, 3π2critical points so, minimum at θ=π12,5π12λ1+λ2=112+512=612=12