The intuition is based on the behavior of polynomial functions. Assume that function "f" has a maximum at "x"0, the reasoning being similar for a function minimum. If formula_33 is a local maximum then, roughly, there is a (possibly small) neighborhood of formula_8 such as the function "is increasing before" and "decreasing after" formula_8. As the derivative is positive for an increasing function and negative for a decreasing function, formula_2 is positive before and negative after formula_8. formula_2 doesn't skip values (by Darboux's theorem), so it has to be zero at some point between the positive and negative values. The only point in the neighbourhood where it is possible to have formula_39 is formula_8.