7. A function f is has the following derivatives (defined for all real numbers x): f'(x) = 3r – 3r and f"(x) = 92? – 3 (a) On what intervals is f increasing? (Use interval notation to express your answer) (b) List all z values at which f has a local maximum. (c) List all z values at which f has a local minimum. (d) On what intervals is f concave up? (e) List all z values at which f has an inflection point. (f) Use the derivatives of f to draw a very rough sketch giving the basic shape of f (label the x-axis in your sketch clearly; you do not have to label the y-axis) Note: your answer must consist of just ONE graph!
7. A function f is has the following derivatives (defined for all real numbers x): f'(x) = 3r – 3r and f"(x) = 92? – 3 (a) On what intervals is f increasing? (Use interval notation to express your answer) (b) List all z values at which f has a local maximum. (c) List all z values at which f has a local minimum. (d) On what intervals is f concave up? (e) List all z values at which f has an inflection point. (f) Use the derivatives of f to draw a very rough sketch giving the basic shape of f (label the x-axis in your sketch clearly; you do not have to label the y-axis) Note: your answer must consist of just ONE graph!
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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![**Exercise 7: Derivatives and Graph Analysis**
A function \( f \) has the following derivatives (defined for all real numbers \( x \)):
\[ f'(x) = 3x^2 - 3x \]
\[ f''(x) = 9x - 3 \]
**Tasks:**
(a) On what intervals is \( f \) increasing? (Use interval notation to express your answer)
(b) List all \( x \) values at which \( f \) has a local maximum.
(c) List all \( x \) values at which \( f \) has a local minimum.
(d) On what intervals is \( f \) concave up?
(e) List all \( x \) values at which \( f \) has an inflection point.
(f) Use the derivatives of \( f \) to draw a very rough sketch giving the basic shape of \( f \). Label the x-axis in your sketch clearly; you do not have to label the y-axis.
**Note:** Your answer must consist of just ONE graph!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F108ede44-c850-4a94-924c-e890d79e0463%2Fbe08b42c-8a9c-4dc0-940c-c51f1dbe5622%2F4isqqck_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 7: Derivatives and Graph Analysis**
A function \( f \) has the following derivatives (defined for all real numbers \( x \)):
\[ f'(x) = 3x^2 - 3x \]
\[ f''(x) = 9x - 3 \]
**Tasks:**
(a) On what intervals is \( f \) increasing? (Use interval notation to express your answer)
(b) List all \( x \) values at which \( f \) has a local maximum.
(c) List all \( x \) values at which \( f \) has a local minimum.
(d) On what intervals is \( f \) concave up?
(e) List all \( x \) values at which \( f \) has an inflection point.
(f) Use the derivatives of \( f \) to draw a very rough sketch giving the basic shape of \( f \). Label the x-axis in your sketch clearly; you do not have to label the y-axis.
**Note:** Your answer must consist of just ONE graph!
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