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18.014ESG Problem Set 5 Pramod N. Achar

Summary: 18.014­ESG Problem Set 5
Pramod N. Achar
Fall 1999
1. (Brouwer Fixed-Point Theorem in dimension 1) Let B1
denote the closed
interval [-1, 1]. Show that if f : B1
is continuous, then there is
some point x0 B1
such that f(x0) = x0. (Such a point is called a fixed
point of f.) (Hint: Define a function g : B1
R by g(x) = f(x) - x, and
apply Bolzano's Theorem to it. Note that the codomain of f is just B1
not R.)
2. Let P be a polynomial of odd degree. Show that P has at least one real
root. (Hint: By Bolzano's Theorem, it suffices to find two points c, d R
such that P(c) 0 and P(d) 0. Let us write P(x) = anxn
+· · ·+a1x+a0,


Source: Achar, Pramod - Department of Mathematics, Louisiana State University


Collections: Mathematics