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-a+ito + DNE - 十→f -2 1 DNE + DNE - -1 DNE DNE | -1 1 2.

-a+ito + DNE - 十→f -2 1 DNE + DNE - -1 DNE DNE | -1 1 2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Sketch a possible graph of f. 
**Transcription for Educational Website** --- ### Problem 62 Explanation The image depicts three number lines associated with a function \( f \), its first derivative \( f' \), and its second derivative \( f'' \). These number lines provide information about the intervals of increase and decrease as well as where the derivatives do not exist (DNE). #### Number Line Details: 1. **Top Line - Function \( f \):** - Represents the behavior of the function \( f \) across the interval. - Negative on the interval \( (-\infty, 0) \). - Positive on the interval \( (0, 1) \). - DNE at \( x = 1 \). - Negative on the interval \( (1, 2) \). - Positive beyond 2. 2. **Middle Line - First Derivative \( f' \):** - Indicates where the function \( f \) is increasing or decreasing. - Positive on the interval \( (-\infty, -1) \). - DNE at \( x = -1 \). - Negative on the interval \( (-1, 0) \). - Positive on the interval \( (0, 1) \). - DNE at \( x = 1 \). - Positive beyond 1. 3. **Bottom Line - Second Derivative \( f'' \):** - Shows the concavity and points of inflection of \( f \). - Positive on the interval \( (-\infty, -1) \). - DNE at \( x = -1 \). - Positive on the interval \( (-1, 1) \). - DNE at \( x = 1 \). - Negative beyond 1. #### Instructions Below the Graphs: - The instructions advise students to sketch careful, labeled graphs of each function \( f \) for given exercises, without using a calculator or graphing utility. - As part of the exercise, students are asked to examine the roots and create sign charts for the given intervals. --- This transcription aids in understanding the relationships between a function and its derivatives. It guides students in visualizing the characteristics of functions through sign charts.
Transcribed Image Text:**Transcription for Educational Website** --- ### Problem 62 Explanation The image depicts three number lines associated with a function \( f \), its first derivative \( f' \), and its second derivative \( f'' \). These number lines provide information about the intervals of increase and decrease as well as where the derivatives do not exist (DNE). #### Number Line Details: 1. **Top Line - Function \( f \):** - Represents the behavior of the function \( f \) across the interval. - Negative on the interval \( (-\infty, 0) \). - Positive on the interval \( (0, 1) \). - DNE at \( x = 1 \). - Negative on the interval \( (1, 2) \). - Positive beyond 2. 2. **Middle Line - First Derivative \( f' \):** - Indicates where the function \( f \) is increasing or decreasing. - Positive on the interval \( (-\infty, -1) \). - DNE at \( x = -1 \). - Negative on the interval \( (-1, 0) \). - Positive on the interval \( (0, 1) \). - DNE at \( x = 1 \). - Positive beyond 1. 3. **Bottom Line - Second Derivative \( f'' \):** - Shows the concavity and points of inflection of \( f \). - Positive on the interval \( (-\infty, -1) \). - DNE at \( x = -1 \). - Positive on the interval \( (-1, 1) \). - DNE at \( x = 1 \). - Negative beyond 1. #### Instructions Below the Graphs: - The instructions advise students to sketch careful, labeled graphs of each function \( f \) for given exercises, without using a calculator or graphing utility. - As part of the exercise, students are asked to examine the roots and create sign charts for the given intervals. --- This transcription aids in understanding the relationships between a function and its derivatives. It guides students in visualizing the characteristics of functions through sign charts.
Expert Solution
Solution:

The function f is increasing on some interval, if f'>0 on that interval.   

The function f is decreasing on some interval, if f'<0 on that interval.

The function f is concave up on some interval, if f''>0 on that interval.   

The function f is concave down on some interval, if f''<0 on that interval. 

On the intervals -∞,-1 and  -1,1, f''>0. Hence, the function f is concave up on these interval.

On the interval 1,∞, f''<0. Hence, the function f is concave down on this interval.

On the intervals -∞,-1 , 0,1 and 1,∞, f'>0. Hence, the function f is increasing  on these interval.

On the interval -1,0, f'<0. Hence, the function f is decreasing on this interval.

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