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11. Let f be a f maximum at I. f'(c II. If f' III. If f" (А) П only

11. Let f be a f maximum at I. f'(c II. If f' III. If f" (А) П only

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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### Mathematical Concepts on Relative Maximum **Question:** Let \( f \) be a function defined and continuous on the closed interval \([a, b]\). If \( f \) has a relative maximum at \( c \) and \( a < c < b \), which of the following statements must be true? I. \( f'(c) \) exists II. If \( f'(c) \) exists, then \( f'(c) = 0 \). III. If \( f'(c) \) exists, then \( f'(c) \leq 0 \). **Options:** - (A) I only - (B) II only - (C) I and II only - (D) I and III only - (E) II and III only **Solution Explanation:** - The problem asks which statements about the function \( f \) are necessarily true when \( f \) has a relative maximum at a point \( c \) within an interval \((a, b)\). - Statement II highlights the necessity of a derivative being zero at a relative maximum if it exists, aligning with the concept of critical points. - The correct answer, marked on the sheet, is option (C): I and II only. This problem consolidates understanding of relative extrema and the role of derivatives in determining such points in calculus.
Transcribed Image Text:### Mathematical Concepts on Relative Maximum **Question:** Let \( f \) be a function defined and continuous on the closed interval \([a, b]\). If \( f \) has a relative maximum at \( c \) and \( a < c < b \), which of the following statements must be true? I. \( f'(c) \) exists II. If \( f'(c) \) exists, then \( f'(c) = 0 \). III. If \( f'(c) \) exists, then \( f'(c) \leq 0 \). **Options:** - (A) I only - (B) II only - (C) I and II only - (D) I and III only - (E) II and III only **Solution Explanation:** - The problem asks which statements about the function \( f \) are necessarily true when \( f \) has a relative maximum at a point \( c \) within an interval \((a, b)\). - Statement II highlights the necessity of a derivative being zero at a relative maximum if it exists, aligning with the concept of critical points. - The correct answer, marked on the sheet, is option (C): I and II only. This problem consolidates understanding of relative extrema and the role of derivatives in determining such points in calculus.
Expert Solution
Step 1

Given :

f has relative maximum at x=c

Suppose f is an absolute function

then f has relative maximum but f'(x) does not exists at corner points

So f'(c) exists is not true

 

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