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0. Given the following functions, make a sign sign diagram for • the first and second derivative and find all the critical points, inflection points, local mins and maximums, intervals of increase and decrease, concavity intervals and use them to sketch the graph of the function : 2 (. f(re) = 27e²³ - 37e² - 36x + 28. 3

0. Given the following functions, make a sign sign diagram for • the first and second derivative and find all the critical points, inflection points, local mins and maximums, intervals of increase and decrease, concavity intervals and use them to sketch the graph of the function : 2 (. f(re) = 27e²³ - 37e² - 36x + 28. 3

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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please help answer all of these calculus qestion. Thank you,

# Calculus Homework Problems ### Problem 1 **Given the following functions, make a sign diagram for the first and second derivative and find all the critical points, inflection points, local minimums and maximums, intervals of increase and decrease, concavity intervals and use them to sketch the graph of the function.** 1. \( f(x) = 9x^3 - 36x^2 - 36x + 28 \) 2. \( f(x) = x^2 (4 - x^2)^2 \) ### Problem 2 **Sketch the graph of a continuous function \( f \) so that:** 1. \( f(2) = 1 \) - \( f'(2) = 0 \), the point (2,1) is a local minimum of \( f \) 2. \( f \) is undefined at \( x = 3 \) - \( f'(x) < 0 \) on \( (-\infty, 3) \cup (3, \infty) \) - \( f''(x) < 0 \) on \( (-\infty, 3) \) - \( f''(x) > 0 \) on \( (3, \infty) \) ### Detailed Explanation of the Process: 1. **Finding the First Derivative**: - For the function \( f(x) \), find \( f'(x) \) which represents the rate of change of the function. 2. **Finding the Second Derivative**: - Find \( f''(x) \) from \( f'(x) \) to understand the concavity of the function. 3. **Critical Points**: - Determine where \( f'(x) = 0 \) or \( f'(x) \) does not exist. These points indicate potential local maximums or minimums. 4. **Inflection Points**: - Find where \( f''(x) = 0 \) or \( f''(x) \) does not exist. These points indicate where the concavity changes from concave up to concave down or vice versa. 5. **Intervals of Increase and Decrease**: - Analyze the sign of \( f'(x) \) to determine intervals where the function is increasing or decreasing. 6. **Concavity Intervals**:
Transcribed Image Text:# Calculus Homework Problems ### Problem 1 **Given the following functions, make a sign diagram for the first and second derivative and find all the critical points, inflection points, local minimums and maximums, intervals of increase and decrease, concavity intervals and use them to sketch the graph of the function.** 1. \( f(x) = 9x^3 - 36x^2 - 36x + 28 \) 2. \( f(x) = x^2 (4 - x^2)^2 \) ### Problem 2 **Sketch the graph of a continuous function \( f \) so that:** 1. \( f(2) = 1 \) - \( f'(2) = 0 \), the point (2,1) is a local minimum of \( f \) 2. \( f \) is undefined at \( x = 3 \) - \( f'(x) < 0 \) on \( (-\infty, 3) \cup (3, \infty) \) - \( f''(x) < 0 \) on \( (-\infty, 3) \) - \( f''(x) > 0 \) on \( (3, \infty) \) ### Detailed Explanation of the Process: 1. **Finding the First Derivative**: - For the function \( f(x) \), find \( f'(x) \) which represents the rate of change of the function. 2. **Finding the Second Derivative**: - Find \( f''(x) \) from \( f'(x) \) to understand the concavity of the function. 3. **Critical Points**: - Determine where \( f'(x) = 0 \) or \( f'(x) \) does not exist. These points indicate potential local maximums or minimums. 4. **Inflection Points**: - Find where \( f''(x) = 0 \) or \( f''(x) \) does not exist. These points indicate where the concavity changes from concave up to concave down or vice versa. 5. **Intervals of Increase and Decrease**: - Analyze the sign of \( f'(x) \) to determine intervals where the function is increasing or decreasing. 6. **Concavity Intervals**:
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