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%I #38 Aug 21 2023 12:19:07
%S 3,8,4,9,0,0,1,7,9,4,5,9,7,5,0,5,0,9,6,7,2,7,6,5,8,5,3,6,6,7,9,7,1,6,
%T 3,7,0,9,8,4,0,1,1,6,7,5,1,3,4,1,7,9,1,7,3,4,5,7,3,4,8,8,4,3,2,2,6,5,
%U 1,7,8,1,5,3,5,2,8,8,8,9,7,1,2,9,1,4,3,5,9,7,0,5,7,1,6,6,3,5,0,1,5,0,1,3,9
%N Decimal expansion of 2/(3*sqrt(3)) = 2*sqrt(3)/9.
%C Consider any cubic polynomial f(x) = a(x - r)(x - (r + s))(x -(r + 2s)), where a, r, and s are real numbers with s > 0 and nonzero a; i.e., any cubic polynomial with three distinct real roots, of which the middle root, r + s, is equidistant (with distance s) from the other two. Then the absolute value of f's local extrema is |a|*s^3*(2*sqrt(3)/9). They occur at x = r + s +- s*(sqrt(3)/3), with the local maximum, M, at r + s - s*sqrt(3)/3 when a is positive and at r + s + s*sqrt(3)/3 when a is negative (and the local minimum, m, vice versa). Of course m = -M < 0.
%C A quadratic number with denominator 9 and minimal polynomial 27x^2 - 4. - _Charles R Greathouse IV_, Apr 21 2016
%C This constant is also the maximum curvature of the exponential curve, occurring at the point M of coordinates [x_M = -log(2)/2 = (-1/10)*A016655; y_M = sqrt(2)/2 = A010503]. The corresponding minimum radius of curvature is (3*sqrt(3))/2 = A104956 (see the reference Eric Billault and the link MathStackExchange). - _Bernard Schott_, Feb 02 2020
%D Eric Billault, Walter Damin, Robert Ferréol et al., MPSI - Classes Prépas, Khôlles de Maths, Ellipses, 2012, exercice 17.07 pages 386, 389-390.
%H MathStackExchange, <a href="https://math.stackexchange.com/questions/1290883/apex-of-an-exponential-function">Apex of an Exponential Function</a>, 2015.
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F (2/9)*sqrt(3) = (2/9)*A002194.
%e 0.384900179459750509672765853667971637098401167513417917345734...
%t RealDigits[2/(3*Sqrt[3]), 10, 105] (* _T. D. Noe_, May 31 2012 *)
%o (PARI) default(realprecision, 1000); 2*sqrt(3)/9
%Y Cf. A002194, A104956, A020760.
%Y Cf. A010503, A016655.
%K nonn,cons
%O 0,1
%A _Rick L. Shepherd_, May 29 2012
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