2. The polynomial $3 x^{5}-250 x^{3}+735 x$ is interesting because it has the maximum possible number of relative extrema and points of inflection at integer lattice points for a quintic polynomial. What is the sum of the $x$-coordinates of these points?
Solution: The first derivative is $15 x^{4}-750 x^{2}+735$, whose roots (which give the relative extrema) sum to $750 / 15=50$. The second derivative is $60 x^{3}-1500 x$, whose roots (which give the points of inflection) sum to $1500 / 60=25$, for a grand total of 75 .