91Ó°ÊÓ

In Exercises \(81-86,\) you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$ f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25] $$

Step by step solution

01

Plot the Function

Using a computer algebra system (CAS) or graphing calculator, plot the function \( f(x) = x^4 - 8x^2 + 4x + 2 \) over the interval \( \left [-\frac{20}{25}, \frac{64}{25} \right ] \). This will help us visualize the behavior of the function and locate potential extrema.

02

Find Critical Points where \( f'(x) = 0 \)

First, find the derivative of \( f(x) \); \( f'(x) = 4x^3 - 16x + 4 \). Then solve \( 4x^3 - 16x + 4 = 0 \) within the interval \( \left [-\frac{20}{25}, \frac{64}{25} \right ] \). Utilize a CAS to find these solutions as they may be complex to solve analytically.

03

Determine Points where \( f'(x) \) Does Not Exist

Examine \( f'(x) = 4x^3 - 16x + 4 \). This is a polynomial, and thus defined for all real \( x \). Therefore, there are no points in the interval where \( f'(x) \) does not exist.

04

Evaluate \( f(x) \) at All Important Points

Evaluate \( f(x) \) at the critical points found in Step 2 and at the endpoints of the interval \( x = -\frac{20}{25} \) and \( x = \frac{64}{25} \). Calculate and compare these values to find potential extrema.

05

Identify Absolute Extrema

Compare the values obtained from evaluating \( f(x) \) at each of the important points. The largest value will be the absolute maximum and the smallest value will be the absolute minimum over the given interval.

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.

Understanding Absolute Extrema

When you're trying to find absolute extrema of a function, you're essentially pinpointing the highest and lowest values the function takes on over a specified interval. These are known as the **absolute maximum** and **absolute minimum**, respectively. To effectively find these extrema, the process involves evaluating the function at certain key points:

• Endpoints of the interval: These are the boundaries of the region you’re analyzing.
• Critical points: These are points where the derivative is zero or does not exist (though in polynomial functions, the derivative generally exists everywhere).

By comparing the values of the function at these critical points and endpoints, you determine where the function reaches its peak and lowest trough within the interval. This is a crucial part of calculus that helps not only in pure math but also in many applied fields to ascertain optimal performance, cost, resource utilization, etc.

Identifying Critical Points

Critical points are vital in finding where a function changes its rate of increase or decrease. Understanding critical points begins with analyzing the derivative of the function. The derivative, denoted as \( f'(x) \), reveals how the function **slopes** or **changes**.

• A critical point occurs where \( f'(x) = 0 \), meaning a horizontal tangent line, suggesting a potential peak or trough (the maximum or minimum).
• It can also be where the derivative is undefined, although for polynomials like in this exercise, the derivative exists everywhere.

To locate these points, solve the equation \( f'(x) = 0 \). The solutions within your specified interval are your critical points. Once identified, these points are part of assessing the function's behavior within the interval, aiding in finding the absolute extrema.

Performing Derivative Analysis

Derivative analysis is an investigation into how a function behaves by examining its derivative. The derivative tells us the rate at which the function's value is changing. It helps in understanding:

• The general trend of the function—whether it's increasing or decreasing.
• Where the function has peaks, valleys, or flat spots, which correspond to critical points.

By taking the derivative, such as in the exercise where \( f'(x) = 4x^3 - 16x + 4 \), you gain insights into these behavior patterns. Solving \( f'(x) = 0 \) helps locate changes in this behavior. This is the cornerstone of calculus as it reveals changes in dynamics, critical for optimizing and understanding real-world scenarios. Derivative analysis, hence, provides a snapshot of how a system modeled by a function might evolve over time or space.