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(a) Letƒ be differentiable on (a, b). Prove that if f'(x) #0 for each x e (a, b), then ƒ has at most one zero in (a, b). (b) Let ƒ be twice differentiable on (a, b). Prove that if f"(x)#0 for each x € (a, b), then ƒ has at most two zeros in (a, b).

(a) Letƒ be differentiable on (a, b). Prove that if f'(x) #0 for each x e (a, b), then ƒ has at most one zero in (a, b). (b) Let ƒ be twice differentiable on (a, b). Prove that if f"(x)#0 for each x € (a, b), then ƒ has at most two zeros in (a, b).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
(a) Let \( f \) be differentiable on \( (a, b) \). Prove that if \( f'(x) \neq 0 \) for each \( x \in (a, b) \), then \( f \) has at most one zero in \( (a, b) \). (b) Let \( f \) be twice differentiable on \( (a, b) \). Prove that if \( f''(x) \neq 0 \) for each \( x \in (a, b) \), then \( f \) has at most two zeros in \( (a, b) \).
Transcribed Image Text:(a) Let \( f \) be differentiable on \( (a, b) \). Prove that if \( f'(x) \neq 0 \) for each \( x \in (a, b) \), then \( f \) has at most one zero in \( (a, b) \). (b) Let \( f \) be twice differentiable on \( (a, b) \). Prove that if \( f''(x) \neq 0 \) for each \( x \in (a, b) \), then \( f \) has at most two zeros in \( (a, b) \).
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