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Chapter 2: Problem 111

Consider the piecewise-defined function \(f\) defined by: \(f(x)=\left\\{\begin{array}{ll}x^{2}+2, & \text { for }-2 \leq x \leq 0 \\ 2, & \text { for } 0

Short Answer

Expert verified
The absolute maximum is 6 at \( x = -2\). The absolute minimum is 2.

Step by step solution

01

Understand the Function Definition

The function is defined in three different intervals. For \( -2 \leq x \leq 0 \), \( f(x) = x^2 + 2 \); for \( 0 < x < 4 \), \( f(x) = 2 \); and for \( 4 \leq x \leq 6 \), \( f(x) = x - 2 \).
02

Sketch Each Segment of the Graph

1. For \( -2 \leq x \leq 0 \), plot the quadratic function \( x^2 + 2 \). This segment is a part of the parabola with vertex at (0, 2).\2. For \( 0 < x < 4 \), plot the constant function \( f(x) = 2 \), which is a horizontal line. This line does not include the endpoints.\3. For \( 4 \leq x \leq 6 \), plot the linear function \( f(x) = x - 2 \). This is a line starting from (4, 2) to (6, 4).
03

Identify the Absolute Maximum

Evaluate the endpoints and any continuous segments:\- At \( x = -2 \), \( f(-2) = (-2)^2 + 2 = 6 \).\- At \( x = 4 \), \( f(4) = 4 - 2 = 2 \).\- At \( x = 6 \), \( f(6) = 6 - 2 = 4 \).\The maximum value is at \( f(-2) = 6 \).
04

Describe the Absolute Minimum

The minimum value can be found by comparing all segments:\- The quadratic part, \( f(x) = x^2 + 2 \), has a minimum value of 2 when \( x = 0 \).\- The constant part, \( f(x) = 2 \), also equals 2 on \( 0 < x < 4 \).\- The linear part, \( f(x) = x - 2 \), reaches its minimum value at \( x = 4 \) which is 2.\Thus, the absolute minimum value is 2.

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Key Concepts

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

Graphing Piecewise Functions
A piecewise-defined function is one that has different expressions based on the input value. For example, consider the function provided: \[f(x) = \begin{cases} x^2 + 2, & \text{for } -2 \leq x \leq 0 \ 2, & \text{for } 0 < x < 4 \ x - 2, & \text{for } 4 \leq x \leq 6 \end{cases} \] To graph such a function:
  • Identify the distinct intervals.
  • Plot each section of the function accordingly.
In this case,
  • For \(-2 \leq x \leq 0\), the graph is a part of a parabola \(y = x^2 + 2\).
  • For \(0 < x < 4\), the function is simply \(y = 2\), which is a horizontal line.
  • From \(4 \leq x \leq 6\), the function becomes linear with \(y = x - 2\).
Ensure to clearly include open and closed dots to represent included and excluded endpoints.
Absolute Maximum
To determine the absolute maximum of a piecewise function, evaluate the function at critical points and endpoints within the domain. For the provided function, evaluate:
  • At \(x = -2\), \[f(-2) = (-2)^2 + 2 = 6\]
  • At \(x = 4\), \[f(4) = 4 - 2 = 2\]
  • At \(x = 6\), \[f(6) = 6 - 2 = 4\]
Compare these values. The highest value is the absolute maximum: \[f(-2) = 6\]. This means the absolute maximum occurs at \(x = -2\) with a value of 6.
Absolute Minimum
For the absolute minimum, you must consider the function values across its entire domain. For each segment of the given function:
  • Over the interval \(-2 \leq x \leq 0\), the minimum of \[f(x) = x^2 + 2\] occurs at \(x = 0\) giving \[f(0) = 2\].
  • Over \(0 < x < 4\), the constant function \[f(x) = 2\] maintains the value of 2.
  • From \(4 \leq x \leq 6\), the function \[f(x) = x - 2\] at \(x = 4\) is also 2.
Thus, the smallest value occurs across all parts of the function, establishing the absolute minimum value as \[f(x) = 2\].
Function Evaluation
Function evaluation involves substituting specific values for \(x\) into the function \(f(x)\) and solving. For piecewise functions:
  • Identify the applicable sub-function based on the \(x\) value.
  • Perform the substitution and simplify.
For example, to evaluate \(f(5)\) in the given piecewise function, determine which interval \(5\) lies in (which is \(4 \leq x \leq 6\)), and apply the relevant rule: \[f(5) = 5 - 2 = 3\]By accurately evaluating each segment accordingly, ensure precise computation across all pieces of the function.

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Most popular questions from this chapter

Graph each function over the given interval. Visually estimate where absolute maximum and minimum values occur. Then use the TABLE feature to refine your estimate. $$f(x)=x^{2 / 3}(x-5) ; \quad[1,4]$$

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Find each limit, if it exists. $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{x^{2}-1}$$

Differentiate implicitly to find \(d^{2} y / d x^{2}\). $$x y+x-2 y=4$$

The percentage of the U.S. civilian labor force aged \(35-44\) may be modeled by \(f(x)=-0.029 x^{2}+0.928 x+19.103\) where \(x\) is the number of years since \(1980 .\) (Source: U.S. Census Bureau.) According to this model, in what year from 1980 through 2010 was this percentage a maximum? Calculate the answer, and then check it on the graph. (GRAPH CAN'T COPY)
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