I was just reading about this interesting question, that Erich's post reminded me of: Call a real polynomial P(x) "tame" if P(0) = 0. I.e., the constant term = 0. Given n (distinct) tame polynomials P_k(x), 1 <= k <= n, we can assume they're numbered such that for all negative x sufficiently near 0, we have P_1(x) > P_2(x) > . . . > P_n(x). Then there exists a unique permutation s in the symmetric group S_n such that for all positive x sufficiently near 0, we have P_s(1)(x) > P_s(2)(x) > . . . > P_s(n)(x) The question is: Are all permutations in S_n realizable by a judicious choice of the n polynomials? --Dan Erich Friedman wrote: ----- consider the problem of which monic 4th degree polynomials x^4 + a x^3 + b x^2 + c x + d have the property that some line is tangent to the graph at 2 different places. . . . . . . -----