KGS math club/solution 5 1
Proof:
Let n be the degree of the polynomial P. Since P has nonzero degree, n>0. The Taylor polynomial of P about the point a is
.
The Taylor polynomial of P about the point b is
.
Note that for all real numbers x, because
. Also, since
for all i, we can rewrite G as
.
Now suppose that . Let
. Then c is greater than zero. Let
such that
. Then
; since P is a nonconstant polynomial this implies that there is a turning point in the interval (z,y). Hence, between any two real numbers x,y with
, there exists a turning point. So P has infinitely many turning points. This is a contradiction, since a polynomial only has finitely many turning points. Therefore,
.
Alternative Proof:
Derivation decreases the degree of a polynomial by 1. Hence, the th derivative of
is a linear polynomial of the form
for some
. By assumption
, hence the graph of this linear polynomial is not horizontal. The requirement
then implies
.