Maximum and Minimum
At higher n, the amount of variation in Bn(x) between x = 0 and x = 1 gets large. For instance,
which shows that the value at x = 0 (and at x = 1) is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. D.H. Lehmer showed that the maximum value of Bn(x) between 0 and 1 obeys
unless n is 2 modulo 4, in which case
(where is the Riemann zeta function), while the minimum obeys
unless n is 0 modulo 4, in which case
These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.
Read more about this topic: Bernoulli Polynomials
Famous quotes containing the words maximum and/or minimum:
“Only at his maximum does an individual surpass all his derivative elements, and become purely himself. And most people never get there. In his own pure individuality a man surpasses his father and mother, and is utterly unknown to them.”
—D.H. (David Herbert)
“After decades of unappreciated drudgery, American women just don’t do housework any more—that is, beyond the minimum that is required in order to clear a path from the bedroom to the front door so they can get off to work in the mourning.”
—Barbara Ehrenreich (20th century)