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Chapter 2: Problem 20

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. TABLE 1 Values of the First Two Derivatives of a Function $$ \begin{array}{ccc} \boldsymbol{x} & \boldsymbol{f}^{\prime}(\boldsymbol{x}) & \boldsymbol{f}^{\prime \prime}(\boldsymbol{x}) \\ \hline 0 \leq x<2 & \text { Positive } & \text { Negative } \\ 2 & 0 & \text { Negative } \\ 2

Short Answer

Expert verified
Relative maximum at x = 2; Inflection point at x = 3.

Step by step solution

01

Title - Find Critical Points

Critical points occur where the first derivative is zero or undefined. From the table, observe the values of f'(x): - f'(2)=0 - f'(4)=0 Thus, the critical points are x = 2 and x = 4.
02

Title - Determine Relative Extremes

Evaluate the second derivative at the critical points to determine the nature of each critical point. - At x = 2, f''(2) is negative, indicating a local maximum. - At x = 4, f''(4) is zero. Since f'(x) does not change signs around x=4, there is no relative extreme at x = 4.
03

Title - Find Inflection Points

Inflection points occur where the second derivative changes sign. From the table:- Between x = 2 and x = 3, f''(x) is negative. - At x = 3, f''(x) changes from negative to positive indicating an inflection point. Therefore, the inflection point is at x = 3.

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Key Concepts

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

First Derivative
The first derivative, denoted as \(f'(x)\), measures the rate at which a function \(f(x)\) changes at any given point. It's the slope of the tangent line to the function's curve. The first derivative tells us whether the function is increasing or decreasing:
  • Positive \(f'(x)\) indicates the function \(f(x)\) is increasing.
  • Negative \(f'(x)\) indicates the function \(f(x)\) is decreasing.
  • Zero \(f'(x)\) indicates a potential critical point鈥攅ither a maximum, minimum, or a flat point.
In the provided table,
The values of \(f'(x)\) help us identify where the function changes its rate of increase or decrease.
For example, \(f'(2) = 0\) and \(f'(4) = 0\) give critical points at \(x = 2\) and \(x = 4\).
Second Derivative
The second derivative, denoted as \(f''(x)\), measures the rate at which the first derivative changes. It provides information on the concavity of the function:
  • Positive \(f''(x)\) indicates the function is concave up (shaped like a U).
  • Negative \(f''(x)\) indicates the function is concave down (shaped like an upside-down U).
  • Zero \(f''(x)\) might indicate a possible inflection point, where the concavity changes.
For instance, in the table:
- At \(x = 2\), \(f''(x)\) is negative, informing us that \(f(x)\) is concave down, suggesting a local maximum at this point.
- At \(x = 4\), \(f''(x)\) is zero, which doesn't suffice alone to conclude an inflection point without more context.
Critical Points
Critical points of a function are points where the first derivative \(f'(x)\) is zero or undefined. These points are crucial because they could be locations of relative maxima, minima, or saddle points. In our case:
  • \(x = 2\): The first derivative \(f'(2) = 0\). This is a critical point.
  • \(x = 4\): The first derivative \(f'(4) = 0\). This, too, is a critical point.
We then use the second derivative to determine the nature of these critical points (maximum, minimum, or neither).
Relative Extremum
To find out what happens at each critical point, we evaluate the second derivative:
  • Local Maximum: If \(f''(x) < 0\) at the critical point, the function has a local maximum there.
  • Local Minimum: If \(f''(x) > 0\) at the critical point, the function has a local minimum there.
  • No Extremum: If \(f''(x) = 0\), further testing is needed since it might be an inflection point rather than an extremum.
From the table:
- At \(x = 2\), \(f''(2)\) is negative, so there is a local maximum at \(x = 2\).
- At \(x = 4\), \(f''(4)\) is zero. Since \(f'(4)\) doesn't change signs around \(x=4\), there isn't a relative extremum at this point.
Inflection Points
Inflection points are points on the curve where the concavity changes. These can be identified by where the second derivative changes sign:
  • If \(f''(x)\) changes from positive to negative or from negative to positive, then \(x\) is an inflection point.
In the provided data:
- Between \(x = 2\) and \(x = 3\), \(f''(x)\) is negative.
- At \(x = 3\), \(f''(x)\) changes from negative to positive. This indicates an inflection point at \(x = 3\).

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