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4 -4 -3 2 4 -2

4 -4 -3 2 4 -2

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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Based on the graph of a function that I attached how do I Find the graph of its derivative 

This image depicts the graph of a cubic polynomial function. The graph is plotted on a Cartesian plane with labeled axes. ### Description - **Axes**: - The horizontal axis is the x-axis, ranging from -5 to 5. - The vertical axis is the y-axis, ranging from -5 to 5. - **Curve**: - The curve intersects the y-axis at approximately \( y = 0 \). - Key points on the graph include a local minimum around \( x = -3 \) and a local maximum around \( x = 1 \). - The function decreases from negative infinity, reaches a local minimum near \( (-3, -4) \), then increases to a local maximum around \( (1, 4) \), before decreasing again. - **Behavior**: - The function exhibits cubic behavior, changing direction twice to form a wave-like shape. - It moves upwards as it approaches the local maximum, then downwards after passing it. This graph can be used to illustrate concepts such as local maxima and minima, points of inflection, and the general shape and behavior of polynomial functions.
Transcribed Image Text:This image depicts the graph of a cubic polynomial function. The graph is plotted on a Cartesian plane with labeled axes. ### Description - **Axes**: - The horizontal axis is the x-axis, ranging from -5 to 5. - The vertical axis is the y-axis, ranging from -5 to 5. - **Curve**: - The curve intersects the y-axis at approximately \( y = 0 \). - Key points on the graph include a local minimum around \( x = -3 \) and a local maximum around \( x = 1 \). - The function decreases from negative infinity, reaches a local minimum near \( (-3, -4) \), then increases to a local maximum around \( (1, 4) \), before decreasing again. - **Behavior**: - The function exhibits cubic behavior, changing direction twice to form a wave-like shape. - It moves upwards as it approaches the local maximum, then downwards after passing it. This graph can be used to illustrate concepts such as local maxima and minima, points of inflection, and the general shape and behavior of polynomial functions.
Expert Solution
Step 1: Have a close look at the graph and write down the information you gather.

See from the given graph of the function fx, the graph becomes parallel to x axis at the points x=-2 and x=1

Thus, the derivative f'x becomes zero at x=-2 and x=1. That is,

f'1=0f'-2=0

Thus, f'x has roots at x=1 and x=-2.

Also see that, fx is an increasing function for -2<x<1. Thus,

f'x>0, for -2<x<1

Also see that, fx is a decreasing  function for -∞<x<-2 and 1<x<∞. Thus,

f'x<0, for -∞<x<-2 and 1<x<∞

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