Nick, you chided
bill, i don't think your claim is accurate. it seems you've found the smallest c such that one edge goes through the origin. but i think one can always find a smaller triangle.
by experimentation, i find that c is suspiciously close to sqrt(3), but i have n o math to support this.
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"R. William Gosper" wrote:
Nick Baxter points out that in answering
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?
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Claim: The midpoint of one triangle side is the origin. Otherwise, centering some side and rotating its endpoints back onto the curve will intersect the ot her two sides with the curve, indicating that c is non-minimal.
NONSENSE! The derivatives at both endpoints are equal, so you can slide them slightly in the same direction so that the other two sides obviously cross the curve, which is impossible when c is minimal. Grinding out dc/dx1=0, dc/dx2=0, 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). Trying for the univariate c polynomial led toward degree 588, but numerically the answer is just c = sqrt(3) ! Erich, is there some way to see this? --rwg