Mathematics — Real Analysis · extreme-and-intermediate-value-theorems
Problem
Suppose g(x) is a monic polynomial of odd degree d, that is,
g(x)=xd+bd−1xd−1+⋯+b1x+b0,
for some real numbers b0,b1,…,bd−1. Show that there exists a K∈N such that g(−K)<0. Hint: Make sure to use the fact that d is odd. You will have to use that (−n)d=−(nd).