1. Consider the function f: R→ R defined by f(x) = x³ − 3x² + 4x + 5 (a) Use intermediate value theorem to show that f has at least one root. (b) Use Rolle's theorem to show that f has at most one root. (So, by (a) and (b), f has exactly one root.) 2. Give an example ƒ : [a, b] → R such that f is differentiable on (a, b), and f(a) = f(b) but there exist no c f'(c) = 0.
1. Consider the function f: R→ R defined by f(x) = x³ − 3x² + 4x + 5 (a) Use intermediate value theorem to show that f has at least one root. (b) Use Rolle's theorem to show that f has at most one root. (So, by (a) and (b), f has exactly one root.) 2. Give an example ƒ : [a, b] → R such that f is differentiable on (a, b), and f(a) = f(b) but there exist no c f'(c) = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. Consider the function f: R→ R defined by f(x) = x³ − 3x² + 4x+5
(a) Use intermediate value theorem to show that f has at least one root.
(b) Use Rolle's theorem to show that f has at most one root.
(So, by (a) and (b), f has exactly one root.)
2. Give an example ƒ : [a, b] → R such that f is differentiable on (a, b), and ƒ(a) = f(b) but there exist no c
f'(c) = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2Ff98f53d0-0746-4533-a470-48fbf4c7bd48%2Fa4xk3er_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Consider the function f: R→ R defined by f(x) = x³ − 3x² + 4x+5
(a) Use intermediate value theorem to show that f has at least one root.
(b) Use Rolle's theorem to show that f has at most one root.
(So, by (a) and (b), f has exactly one root.)
2. Give an example ƒ : [a, b] → R such that f is differentiable on (a, b), and ƒ(a) = f(b) but there exist no c
f'(c) = 0.
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