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Let f (x) = cos² x – 2 sin x. 0 < x < 2a. a. FInd all critical numbers for f (x) in the interval. b. Find all intervals where f is increasing or decreasing c. Find all inflection points.

Let f (x) = cos² x – 2 sin x. 0 < x < 2a. a. FInd all critical numbers for f (x) in the interval. b. Find all intervals where f is increasing or decreasing c. Find all inflection points.

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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Need help with this calc homework. Thanks! 

**Function Analysis** Let \( f(x) = \cos^2 x - 2 \sin x \). The domain is \( 0 \leq x \leq 2\pi \). **a. Find all critical numbers for \( f(x) \) in the interval.** Critical numbers occur where the derivative \( f'(x) \) is zero or undefined. **b. Find all intervals where \( f(x) \) is increasing or decreasing.** Determine the sign of \( f'(x) \) in each interval between critical numbers to identify whether \( f(x) \) is increasing or decreasing. **c. Find all inflection points.** Inflection points occur where the second derivative \( f''(x) \) changes signs. **d. Find all intervals of concavity.** Evaluate the sign of \( f''(x) \) in order to determine where \( f(x) \) is concave up or concave down.
Transcribed Image Text:**Function Analysis** Let \( f(x) = \cos^2 x - 2 \sin x \). The domain is \( 0 \leq x \leq 2\pi \). **a. Find all critical numbers for \( f(x) \) in the interval.** Critical numbers occur where the derivative \( f'(x) \) is zero or undefined. **b. Find all intervals where \( f(x) \) is increasing or decreasing.** Determine the sign of \( f'(x) \) in each interval between critical numbers to identify whether \( f(x) \) is increasing or decreasing. **c. Find all inflection points.** Inflection points occur where the second derivative \( f''(x) \) changes signs. **d. Find all intervals of concavity.** Evaluate the sign of \( f''(x) \) in order to determine where \( f(x) \) is concave up or concave down.
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