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Chapter 2: Problem 121

Graph each function over the given interval. Visually estimate where any absolute extrema occur. Then use the TABLE feature to refine each estimate. $$f(x)=\frac{3}{4}\left(x^{2}-1\right)^{2 / 3} ; \quad\left[\frac{1}{2}, \infty\right)$$

Short Answer

Expert verified
The absolute minimum occurs at \( x = 1 \), \( f(1) = 0 \); no absolute maximum exists.

Step by step solution

01

Understand the Function

The function given is \( f(x) = \frac{3}{4} (x^2 - 1)^{2/3} \). This is a power function with a fractional exponent, which means that it is continuous for all \( x \) where \( x^2 - 1 \geq 0 \). Thus, the function is defined for \( |x| \geq 1 \). Since the provided interval is \( \left[\frac{1}{2}, \infty\right) \), we will focus on the part of the function that starts from \( x = 1 \).
02

Graph the Function

Plot the function \( f(x) = \frac{3}{4} (x^2 - 1)^{2/3} \) using a graphing tool or calculator, starting from \( x = 1 \) as the function is not defined at \( x = \frac{1}{2} \). Observe where the function appears to reach any maximum or minimum values in the given interval \( [1, \, \infty) \).
03

Analyze the Graph

Upon plotting, observe that the function appears to increase from \( x = 1 \) onwards. It does not have a maximum value in the interval since it continues to rise indefinitely.
04

Use the TABLE Feature

Use the TABLE feature on a calculator to evaluate \( f(x) \) at specific points starting from \( x = 1 \). Check values such as \( x = 2, 3, 4 \), and note how \( f(x) \) increases. This confirms there's no maximum in this interval, and the minimum occurs at \( x = 1 \), which corresponds to the value of \( 0 \).
05

Confirm the Extreme Values

The absolute minimum in the interval \([1, \infty)\) is at \( x = 1 \) with \( f(1) = 0 \). There is no absolute maximum because \( f(x) \) increases indefinitely.

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Key Concepts

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

Absolute Extrema
Absolute extrema refer to the highest or lowest points that a function reaches over a particular interval. They are the points at which a function takes its largest or smallest values, called the absolute maximum or minimum, respectively.
To find absolute extrema, follow these steps:
  • Identify critical points where the derivative of the function is zero or undefined.
  • Evaluate the function at these critical points and at the endpoints of the interval.
  • Compare these values to determine the highest and lowest values.
In our exercise with the function \( f(x) = \frac{3}{4} (x^2 - 1)^{2/3} \), we focus on the interval \([1, \infty)\). Since the function is only defined from \( x = 1 \) onwards, our critical examination reveals that \( f(1) = 0 \), making it the absolute minimum. As the function continues to grow without bound, there is no absolute maximum.
Power Function
A power function is a type of function that can be expressed in the form \( f(x) = ax^n \), where \( a \) is a constant, and \( n \) is a real number. In this exercise, the function \( f(x) = \frac{3}{4} (x^2 - 1)^{2/3} \) is a power function with a fractional exponent.
Understanding fractional exponents is crucial here:
  • Fractional exponents like \( (x^2 - 1)^{2/3} \) indicate a combination of root and power operations.
  • The numerator of the fraction (\( 2 \)) represents the power, while the denominator (\( 3 \)) represents the root.
  • This means \( (x^2 - 1)^{2/3} \) is equivalent to \( \sqrt[3]{(x^2 - 1)^2} \).
Power functions are often continuous and differentiable, making them useful for finding extrema. In this graph analysis, recognizing that the function is well-behaved and increasing helps predict its behavior across the interval.
Graphical Analysis
Graphical analysis involves studying a function's graph to visually estimate important characteristics like trends, intercepts, and extrema. It complements numerical and algebraic analysis to give a comprehensive understanding of the function's behavior.
Here is how you can perform graphical analysis:
  • Plot the function graph using a graphing tool, focusing on the defined domain.
  • Look for areas where the function appears to level off or change direction, indicating potential extrema.
  • Use tools like the TABLE feature of calculators to check values at strategic points.
For the exercise function \( f(x) = \frac{3}{4} (x^2 - 1)^{2/3} \), plotting begins at \( x = 1 \) since it is undefined at any \( x < 1 \). Observing the graph tells us the function increases continually without an upper bound, affirming it has no maximum in the given interval, but it does have an absolute minimum at the start, \( x = 1 \).

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Most popular questions from this chapter

The following table gives the pressure of a pregnant woman's contractions as a function of time. $$\begin{array}{|c|c|}\hline \begin{array}{c} \text { TIME, } T \\\\\text { (in minutes) }\end{array} & \begin{array}{c}\text { PRESSURE } \\\\\text { (in millimeters of mercury) }\end{array} \\\\\hline 0 & 10 \\\1 & 8 \\\2 & 9.5 \\\3 & 15 \\\4 & 12 \\\5 & 14 \\\6 & 14.5\\\\\hline\end{array}$$ Use a calculator that has the REGRESSION option. a) Fit a linear equation to the data. Predict the pressure of the contractions after \(7 \mathrm{~min}\) b) Fit a quartic polynomial to the data. Predict the pressure of the contractions after \(7 \mathrm{~min}\). Find the smallest contraction over the interval [0,10]

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Corner Stone Electronics determines that its weekly profit, in dollars, from the production and sale of \(x\) amplifiers is$$P(x)=\frac{1500}{x^{2}-6 x+10}$$ Find the number of amplifiers, \(x\), for which the total weekly profit is a maximum.

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=2 x^{4}+x ; \quad[-1,1]$$
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