4

Part of 1974 Czech and Slovak Olympiad III A

Problems(1)

Uniformly bounded set of polynomials

Source: Czech and Slovak Olympiad 1974, National Round, Problem 4

\mathcal M

be the set of all polynomial functions

f

of degree at most 3 such that

\forall x\in[-1,1]:\ |f(x)|\le 1.

a

the (possibly zero) coefficient of

f

x^3.

Show that there is a positive number

k

\forall f\in\mathcal M:\ |a|\le k

and find the least

k

with this property.

algebrapolynomialfunctionboundedset