a post from Erich Friedman <efriedma@stetson.edu> --rich --- greetings funsters, 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. there are some hard ways to think about this, equating slopes with average slopes, but the solution is obvious once you realize that these are the polynomials with 2 inflection points, so take 2 derivatives and look for 2 real roots. turns out the condition is 8b < 3a^2. the fact that c and d don't matter should be clear because of horizontal and vertical shifts. all that is background for my real question, one i am having trouble answering. which monic 6th degree polynomials x^6 + A x^5 + B x^4 + C x^3 + D x^2 + Ex + F have a line that is tangent to the graph at 3 different places? if such a line y = U x + V exists, then the polynomial x^6 + A x^5 + B x^4 + C x^3 + D x^2 + Ex + F - Ux - V is tangent to the x-axis at 3 points, which means it is of the form (x - r1)^2 * (x - r2)^2 * (x - r3)^2 for some distinct r1, r2, r3, but which A, B, C, and D accomplish that? i can't seem to solve the resulting equations. is there an easier way of doing this? erich friedman