|
|
A198735
|
|
Decimal expansion of the absolute minimum of f(x)+f(2x), where f(x)=sin(x)+cos(x).
|
|
10
|
|
|
2, 0, 9, 2, 9, 8, 6, 7, 8, 0, 2, 5, 1, 1, 0, 3, 4, 2, 5, 9, 2, 2, 6, 7, 8, 2, 0, 1, 3, 6, 9, 7, 6, 0, 4, 3, 2, 3, 7, 0, 2, 4, 5, 6, 1, 8, 8, 4, 1, 8, 6, 1, 5, 1, 3, 2, 4, 6, 3, 7, 7, 7, 7, 1, 1, 4, 9, 9, 2, 3, 4, 3, 2, 3, 2, 1, 6, 3, 6, 4, 5, 2, 0, 8, 2, 5, 7, 4, 5, 9, 9, 1, 4, 5, 2, 4, 9, 9, 5
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
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)
|
|
LINKS
|
|
|
EXAMPLE
|
x=4.89312267296329905539673190581...
min=-2.09298678025110342592267820137...
|
|
MATHEMATICA
|
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 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|