/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Using a calculator or computer, ... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

Using a calculator or computer, graph the functions in Problems \(8-13 .\) Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$f(x)=3 x^{5}-5 x^{3}$$

Short Answer

Expert verified
Critical points at \( x = -1 \), \( x = 0 \), \( x = 1 \); increasing on \((-\infty, -1)\) and \((1, \infty)\); decreasing on \((-1, 0)\) and \((0, 1)\).

Step by step solution

01

Graph the Function

Use a graphing calculator or graphing software to plot the function \( f(x) = 3x^5 - 5x^3 \). Observe the overall shape, the number of turning points, and where the graph increases or decreases.
02

Identify Critical Points

To find the critical points of the function, first take the derivative: \( f'(x) = 15x^4 - 15x^2 \). Set the derivative equal to zero to find the critical points:\[15x^4 - 15x^2 = 0\]Factor out \( 15x^2 \):\[15x^2(x^2 - 1) = 0\]This gives the solutions \( x = 0 \) and \( x = \pm 1 \). These are the critical points.
03

Determine Function Behavior

Evaluate the intervals between the critical points to determine where the function is increasing or decreasing. Use test points in the derivative \( f'(x) \):- For \( x < -1 \): choose \( x = -2 \), \( f'(-2) > 0 \), so increasing.- For \( -1 < x < 0 \): choose \( x = -0.5 \), \( f'(-0.5) < 0 \), so decreasing.- For \( 0 < x < 1 \): choose \( x = 0.5 \), \( f'(0.5) < 0 \), so decreasing.- For \( x > 1 \): choose \( x = 2 \), \( f'(2) > 0 \), so increasing.
04

Analyze Critical Points

Use the first or second derivative test to categorize the critical points. Based on sign changes in \( f'(x) \):- \( x = -1 \) changes from increasing to decreasing, so it's a local maximum.- \( x = 0 \) changes from decreasing to decreasing, so it's a point of inflection.- \( x = 1 \) changes from decreasing to increasing, so it's a local minimum.
05

Summarize Graph Features

The function \( f(x) = 3x^5 - 5x^3 \) has critical points at \( x = -1 \), \( x = 0 \), and \( x = 1 \).- The function is increasing on \( (-\infty, -1) \) and \( (1, \infty) \).- The function is decreasing on \( (-1, 0) \) and \( (0, 1) \).- There is a local maximum at \( x = -1 \) and a local minimum at \( x = 1 \).- The graph has a point of inflection at \( x = 0 \), where the function is neither increasing nor decreasing.

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
In calculus, critical points of a function are the values of the variable where its derivative is zero or undefined. Understanding these points is key because they often reveal important features of the function's graph, such as peaks, valleys, or transitions. For the function \( f(x) = 3x^5 - 5x^3 \), we start by finding the derivative, which is \( f'(x) = 15x^4 - 15x^2 \).
By setting the derivative equal to zero (\( 15x^4 - 15x^2 = 0 \)), and factoring out \( 15x^2 \), we solve for \( x \) to find our critical points: \( x = 0 \) and \( x = \pm 1 \).
These points are essential as they indicate where the slope of the tangent to the function is zero, meaning the function levels out momentarily. Such points often lead to interesting graph features like local minima or maxima, revealing much about the function's behavior around these values.
Monotonic Function
A function is said to be monotonic when it is entirely non-increasing or non-decreasing over its domain interval. However, most functions aren't strictly monotonic everywhere; they switch between increasing and decreasing at critical points. By examining the derivative of a function, we can determine where these changes happen.
For \( f(x) = 3x^5 - 5x^3 \), use the critical points \( x = -1, 0, \text{and} 1 \) to sectionalize the graph into intervals. By employing test points in these intervals:
  • For \( x < -1 \), the function is increasing.
  • Between \( -1 \) and \( 0 \), it's decreasing.
  • Between \( 0 \) and \( 1 \), it continues to decrease.
  • For \( x > 1 \), the function increases again.
Understanding where a function is monotonic helps predict its overall shape and behavior within those intervals, offering valuable insight into the nature of the graph.
Derivative Analysis
Derivative analysis is a method employed in calculus to understand the characteristics of a function. The derivative, \( f'(x) \), provides information on the rate of change of \( f(x) \). By observing where \( f'(x) \) is positive or negative, we can determine intervals where the function is increasing or decreasing.
In our example \( f(x) = 3x^5 - 5x^3 \), the derivative simplifies to \( f'(x) = 15x^4 - 15x^2 \). Simplifying further, we have \( f'(x) = 15x^2(x^2-1) \). By setting \( f'(x) = 0 \), we identify critical points as \( x = -1, 0, \) and \( 1 \).
Using these critical points in conjunction with test points, derivative analysis enables us to map the function's behavior in detail. It’s like having a roadmap of the graph, indicating where the function changes its direction or speed.
Turning Points
Turning points on a graph of a function are the points where the function changes from increasing to decreasing or vice-versa. These are also known as local maxima and minima. To identify them, we rely on critical points data and further analyze using the first or second derivative test.
For the function \( f(x) = 3x^5 - 5x^3 \), analysis of \( f'(x) = 15x^4 - 15x^2 \) showed critical points at \( x = -1, 0, \) and \( 1 \). Using the first derivative test, we can conclude:
  • At \( x = -1 \), the function changes from increasing to decreasing, marking a local maximum.
  • At \( x = 0 \), there's no change in increasing or decreasing behavior, hence it's a point of inflection rather than a turning point.
  • At \( x = 1 \), the function transitions from decreasing to increasing, indicative of a local minimum.
Understanding these turning points allows us to predict and describe the shapes and sections of the function's graph with confidence.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A landscape architect plans to enclose a 3000 square-foot rectangular region in a botanical garden. She will use shrubs costing \(\$ 45\) per foot along three sides and fencing costing \(\$ 20\) per foot along the fourth side. Find the minimum total cost.

A company can produce and sell \(f(L)\) tons of a product per month using \(L\) hours of labor per month. The wage of the workers is \(w\) dollars per hour, and the finished product sells for \(p\) dollars per ton. (a) The function \(f(L)\) is the company's production function. Give the units of \(f(L) .\) What is the practical significance of \(f(1000)=400 ?\) (b) The derivative \(f^{\prime}(L)\) is the company's marginal product of labor. Give the units of \(f^{\prime}(L) .\) What is the practical significance of \(f^{\prime}(1000)=2 ?\) (c) The real wage of the workers is the quantity of product that can be bought with one hour's wages. Show that the real wage is \(w / p\) tons per hour. (d) Show that the monthly profit of the company is $$ \pi(L)=p f(L)-w L $$ (e) Show that when operating at maximum profit, the company's marginal product of labor equals the real wage: $$ f^{\prime}(L)=\frac{w}{p} $$

The sum of two nonnegative numbers is \(100 .\) What is the maximum value of the product of these two numbers?

On the west coast of Canada, crows eat whelks (a shellfish). To open the whelks, the crows drop them from the air onto a rock. If the shell does not smash the first time, the whelk is dropped again. " The average number of drops, \(n,\) needed when the whelk is dropped from a height of \(x\) meters is approximated by $$n(x)=1+\frac{27}{x^{2}}.$$ (a) Give the total vertical distance the crow travels upward to open a whelk as a function of drop height, \(x\) (b) Crows are observed to drop whelks from the height that minimizes the total vertical upward distance traveled per whelk. What is this height?

Find the point where the following curve is steepest: $$y=\frac{50}{1+6 e^{-2 t}} \quad \text { for } t \geq 0$$

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.