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I would like to find the min of the following function,

-5 + 2 Sqrt[5] + 4 (5 - 3 Sqrt[5]) b^2 + Sqrt[2] (-1 + 2 b^2 - 2 a Sqrt[1 - a^2 - 
2 b^2] Cos[ρ - ϛ])

for -1 ⩽ a ⩽ 1 -((1 - a^2)/2)^(1/2) ⩽ b ⩽ ((1 - a^2)/2)^(1/2) 0 ⩽ ρ ⩽ 2*Pi 0 ⩽ ϛ ⩽ 2*Pi

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    $\begingroup$ Seems like the sort of thing one finds in the documentation. $\endgroup$ Feb 9, 2020 at 13:18

1 Answer 1

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f = -5 + 2 Sqrt[5] + 4 (5 - 3 Sqrt[5]) b^2 + Sqrt[2] (-1 + 2 b^2 - 
 2 a Sqrt[1 - a^2 - 2 b^2] Cos[ρ - ϛ])

Minimize[{f, -1 ⩽ a ⩽ 1, -((1 - a^2)/2)^(1/2) ⩽ b ⩽ ((1 - a^2)/2)^(1/2), 
 0 ⩽ ρ ⩽ 2*Pi, 0 ⩽ ϛ ⩽ 2*Pi}, {a, b, ρ, ϛ}]

{5 - 4 Sqrt[5], {a -> 0, b -> -(1/Sqrt[2]), ρ -> 27/22, ϛ -> 1/22 (27 + 11 π)}}

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