91Ó°ÊÓ

The first and second derivatives of the function \(f(x)\) have the values given in Table 1. (a) Find the \(x\) -coordinates of all relative extreme points. (b) Find the \(x\) -coordinates of all inflection points. $$\text { Table 1 Values of the First Two Derivatives of a Function }$$ $$\begin{array}{ccc} \hline x & f^{\prime}(x) & f^{\prime \prime}(x) \\ \hline 0 \leq x < 2 & \text { Positive } & \text { Negative } \\ 2 & 0 & \text { Negative } \\ 2 < x < 3 & \text { Negative } & \text { Negative } \\ 3 & \text { Negative } & 0 \\ 3 < x < 4 & \text { Negative } & \text { Positive } \\ 4 & 0 & 0 \\ 4 < x \leq 6 & \text { Negative } & \text { Negative } \\ \hline \end{array}$$

Step by step solution

01

Identify critical points

Critical points occur where the first derivative is zero or undefined. From Table 1, the points where the first derivative, \(f'(x)\), is zero are at \(x = 2\) and \(x = 4\).

02

Determine the nature of critical points

Use the second derivative to determine if the critical points are relative maxima, minima, or neither. For \(x = 2\): - \(f'(x) = 0\) - \(f''(x) < 0\) (negative) This indicates a relative maximum at \( x = 2 \). For \(x = 4\): - \(f'(x) = 0\) - \(f''(x) = 0\) More information is needed for \(x = 4\); however, since the intervals around \(x = 4\) show consistent negative values of \(f''(x)\), we need to investigate further if no change in concavity occurs.

03

Find inflection points

Inflection points occur where the second derivative changes sign.Look at \( x = 3 \): - \(f''(x)\) changes from negative for \(2 < x < 3\) to positive for \(3 < x < 4\). Thus, there is an inflection point at \(x = 3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points

Critical points of a function are where its first derivative, \( f'(x) \), is zero or undefined. These points can indicate locations of potential local maxima, minima, or saddle points. Identifying critical points involves:

• Finding where \( f'(x) = 0 \) or is undefined.
• Analyzing those points to determine whether they correspond to relative extrema or other types of stationary points.

From the table, we see the first derivative, \( f'(x) \), is zero at \( x = 2 \) and \( x = 4 \). These are our critical points and require further investigation.

Relative Extrema

Relative extrema are the points where a function reaches a local maximum or minimum. To determine if a critical point is a relative extremum, we can use the second derivative test or the first derivative test.

### Second Derivative Test
Using the second derivative, \( f''(x) \), helps classify the nature of each critical point:

• If \( f''(x) > 0 \) at a critical point, that point is a relative minimum.
• If \( f''(x) < 0 \) at a critical point, that point is a relative maximum.
• If \( f''(x) = 0 \) at a critical point, the test is inconclusive.

For \( x = 2 \):

• \( f'(x) = 0 \)
• \( f''(x) < 0 \)
• This indicates a relative maximum at \( x = 2 \).

For \( x = 4 \):

• \( f'(x) = 0 \)
• \( f''(x) = 0 \)
• This result is inconclusive, but further observation might help classify the nature better.

Inflection Points

Inflection points are where the concavity of the function changes. Detecting these points requires analyzing changes in the sign of the second derivative, \( f''(x) \).

An inflection point occurs when:

• \( f''(x) \) changes from positive to negative or vice versa.
• This switch in signs indicates a change in concavity from concave up to concave down, or concave down to concave up.

The table reveals an inflection point at \( x = 3 \):

• For \( 2 < x < 3 \): \( f''(x) \) is negative.
• For \( 3 < x < 4 \): \( f''(x) \) is positive.
• This change in sign illustrates a shift in the concavity of the function so, there's an inflection point at \( x = 3 \).

First Derivative Test

The first derivative test is another method to determine the nature of critical points. It involves looking at the signs of the first derivative \( f'(x) \) before and after the critical points.

Steps for the First Derivative Test:

• Identify the critical points where \( f'(x) = 0 \) or \( f'(x) \) is undefined.
• Examine the sign of \( f'(x) \) on intervals around each critical point.

For a critical point at \( c \):

• If \( f'(x) \) changes from positive to negative as \( x \) increases through \( c \), \( f(c) \) is a relative maximum.
• If \( f'(x) \) changes from negative to positive as \( x \) increases through \( c \), \( f(c) \) is a relative minimum.
• If \( f'(x) \) does not change signs, \( f(c) \) is neither a maximum nor a minimum.

Applying this to our exercise:

For \( x = 2 \), the derivative changes from positive to negative, indicating a relative maximum.
For \( x = 4 \), more investigation is necessary due to the zero second derivative, so the first or alternative tests might be used.