4. (Lecture 14) The following problems are nice applications of the intermediate value theorem. (a) Let f : [0,3] → R be

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4 Lecture 14 The Following Problems Are Nice Applications Of The Intermediate Value Theorem A Let F 0 3 R Be 1 (98.6 KiB) Viewed 33 times

4. (Lecture 14) The following problems are nice applications of the intermediate value theorem. (a) Let f : [0,3] → R be defined by f(x) = x4 – 3x2 – 7x – 3. = = 10 Prove that f has a root in (0, 3). (b) Let g : [0, 1] → [0, 1] be continuous. Prove that g has a fixed point, i.e., a point 20 € [0, 1] such that g(x0) [Hint: Consider the function ğ : [0, 1] → R defined by ſ(x) := g(x) — 2.) (c) Let h : [0, 1] → R be a continuous function satisfying h(0) = h(1). Prove that there are a, b e [0, 1] satisfying la – b] = { such that h(a) = h(b). [Hint: Consider the function ñ : [0, į] → R defined by ñ() = h(x + 7) – h(x).] =