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Decimal expansion of the abscissa of the minimum of log(x)*log(1+x).
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%I #12 Jun 23 2021 09:06:20

%S 3,1,8,3,6,5,7,3,6,9,4,1,0,9,9,1,8,4,0,2,6,1,1,4,3,8,6,2,8,5,7,6,2,8,

%T 4,6,5,8,2,6,7,0,1,5,9,1,4,5,6,8,8,4,3,3,7,4,7,4,3,7,2,6,6,3,1,8,5,4,

%U 9,5,8,7,4,9,2,2,1,3,2,1,0,0,8,4,3,3,8

%N Decimal expansion of the abscissa of the minimum of log(x)*log(1+x).

%C The function f(x)=log(x)*log(1+x) is zero at x=0 and zero at x=1, and attains a local minimum at x=0.318...

%C This constant equals the zero of the first derivative, i.e., it satisfies (1+x)*log(1+x) + x*log(x) = 0.

%e Equals 0.3183657369410991840261143... The ordinate value is -0.31634671137020605439495...

%p Digits := 100 ; fsolve( (1+x)*log(1+x)+x*log(x)) ;

%t RealDigits[x /. FindRoot[(1 + x)*Log[1 + x] + x*Log[x], {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* _Amiram Eldar_, Jun 23 2021 *)

%o (PARI) solve(x=0.3,0.4,x*log(x)+(1+x)*log(1+x)) \\ _M. F. Hasler_, Jun 23 2021

%K nonn,cons

%O 0,1

%A _R. J. Mathar_, Jun 21 2021