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
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