what is the smallest constant c so that the graph of the function f(x) = x^3 - c x contains the vertices of an equilateral triangle?
<My braindamaged symmetry argument.>
NONSENSE! (Mine, not Nick's.)
Grinding out dc/dx1=0, dc/dx2=0,
Maybe better would have been to guess that for the critical c, the triangle would be unique (but for its negative twin). This would show up as a zero discriminant.
the x coordinates in the minimal case are the roots near -.54557 and .83587 of
6 4 2 27 x - 54 sqrt(3) x + 72 x - 8 sqrt(3).
The triangle abscissae are the three roots %pi 2 %pi %pi 2 %pi 2 sin(---) sin(-----) 2 sin(---) cos(-----) 18 9 9 9 [- 2 sqrt(---------------------), 2 sqrt(---------------------), 3 3 %pi %pi 2 cos(---) cos(---) 18 9 2 sqrt(-------------------)] 3 The area is 2, and the center is (sqrt(2)/3^(3/4),sqrt(2)/3^(5/4)). The slope of the side joining the first and third vertices is tan(pi/18). Macsyma users can see the picture by defining xy(x):=[x,x*(x^2-sqrt(3.))] and then executing block([equalscale:true,x1:-0.54557,x2:0.83587,x3], x3:-sqrt(3.)*(x2^3-x1^3)/2+2*x2-x1, paramplot(''([xy(-x3*(1-t)+x3*t),xy(x1)*(1-t)+t*xy(x2),xy(x2)*(1-t)+t*xy(x3), xy(x3)*(1-t)+t*xy(x1)]),t,0,1))
Trying for the univariate c polynomial led toward degree 588, but numerically the answer is just c = sqrt(3) !
Which can then be verified symbolically. Much harder might be to symbolically verify the centroid expression, found with Rich's old number-relator. The elegant results c=sqrt(3), area=2 raise hope that some dazzling insight will obviate my strenuous calculations, but those nontrivial abscissa values lower my anticipation. This looks like a classic case of intermediate expression swell. Erich, did you invent this bruiser? ---------- prod(uct )i(nte)g(r)al $.02 : P They're not terribly fruitful, since you can "do" I f(x) iff you can / / "do" | ln f(x), but sometimes the product limit seems the only way / to get, e.g., the Barnes-Alexeiewsky recurrence x / | ln t! dt = ln sqrt(2 pi) - (1 - ln x) x . / x-1 --rwg