Lemma: Interval of a Function with Simple Roots
Let f be a function with simple roots such that f(c)=0
Then there exists an interval (a,b) such that:
c is in (a,b)
for all x in (a,b), f'(x) is all positive or all negative
Proof:
(1) Since f has only simple roots and f(c)=0, then it follows that f'(c) ≠ 0. [See Corollary 1.1, here]
(2) Since f is a polynomial, we know that f'(x) is continuous. [See Corollary 2.1, here]
(3) Since f'(c) is nonzero, let ε be nonzero and less than abs{ f'(c) }.
(4) Since f'(x) is continuous at c, there exists a number δ such that if x is in the (c - δ, c + δ), then f'(x) is in (f'(c)-ε, f'(c)+ε). [By the definition of a continuous function]
(5) Since ε is less than abs{f'(c) }, it follows that for x in (c -δ, c + δ), f'(x) is either entirely positive or entirely negative.
QED
No comments :
Post a Comment