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A198745
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Decimal expansion of the absolute minimum of f(x)+f(2x), where f(x)=sin(x)-cos(x).
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10
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2, 7, 4, 1, 8, 0, 1, 4, 1, 0, 8, 4, 6, 8, 0, 8, 0, 5, 1, 2, 7, 6, 0, 1, 5, 7, 1, 5, 5, 3, 2, 4, 4, 4, 6, 7, 5, 9, 5, 1, 6, 3, 5, 6, 9, 4, 6, 9, 6, 8, 6, 4, 6, 9, 9, 9, 6, 0, 8, 6, 5, 2, 2, 3, 2, 2, 5, 8, 9, 7, 1, 1, 4, 4, 0, 7, 3, 4, 3, 6, 7, 0, 4, 8, 1, 8, 1, 1, 1, 5, 2, 4, 0, 0, 5, 2, 2, 2, 4
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OFFSET
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1,1
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COMMENTS
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Let f(x)=sin(x)+cos(x) and g(x)=f(x)+f(2x)+...+f(nx), where n>=2. Then f(x) attains an absolute minimum at some x between 0 and 2*pi. Guide to related sequences (including graphs in Mathematica programs):
n....x.........minimum of f(x)
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LINKS
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EXAMPLE
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x=5.81273216913788444549287183000...
min=-2.74180141084680805127601571...
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MATHEMATICA
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f[t_] := Sin[t] + Cos[t]
x = Minimize[f[t] + f[2 t], t]
N[x, 30]
(RealDigits[N[{#1[[1]], t /. #1[[2]]}, 110]] &)[x]
Plot[f[t] + f[2 t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
Root[27 + 162x - 207x^2 - 8x^3 + 32x^4, 1] // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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