/*! 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 7 7-30. For each function: a. Ma... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

7-30. For each function: a. Make a sign diagram for the first derivative. b. Make a sign diagram for the second derivative. c. Sketch the graph by hand, showing all relative extreme points and inflection points. $$ f(x)=x^{3}+3 x^{2}-9 x+5 $$

Short Answer

Expert verified
Local maximum at \( x = -3 \), local minimum at \( x = 1 \), inflection point at \( x = -1 \).

Step by step solution

01

Find the First Derivative

To begin, compute the first derivative of the function \( f(x) = x^3 + 3x^2 - 9x + 5 \). The derivative, \( f'(x) \), is calculated as follows:\[ f'(x) = 3x^2 + 6x - 9 \].
02

Solve for Critical Points

Next, find the critical points by setting the first derivative to zero and solving for \( x \):\[ 3x^2 + 6x - 9 = 0 \]Factor the equation:\[ 3(x^2 + 2x - 3) = 0 \]\[ 3(x+3)(x-1) = 0 \]The critical points are \( x = -3 \) and \( x = 1 \).
03

Create First Derivative Sign Diagram

Using the critical points, create a sign diagram for \( f'(x) \):- Test intervals: \((-\infty, -3), (-3, 1), (1, \infty)\).- Choose test points within these intervals to determine the sign of \( f'(x) \). For instance, test points can be \( x = -4, 0, 2 \).- Compute the sign of \( f'(x) \) at each test point: - \( f'(-4) = 3(-4+3)(-4-1) = 3(-1)(-5) > 0 \) - \( f'(0) = 3(0+3)(0-1) < 0 \) - \( f'(2) = 3(2+3)(2-1) > 0 \)- Understand the sign diagram shows that the function is increasing on \((-\infty, -3) \) and \((1, \infty)\); decreasing on \((-3, 1)\).
04

Find the Second Derivative

Compute the second derivative from the first derivative:\[ f''(x) = 6x + 6 \].
05

Solve for Inflection Points

Set the second derivative equal to zero to find inflection points:\[ 6x + 6 = 0 \]Solve for \( x \):\[ x = -1 \].
06

Create Second Derivative Sign Diagram

Using the inflection point, create a sign diagram for \( f''(x) \):- Test intervals: \((-\infty, -1), (-1, \infty)\).- Choose test points, e.g., \( x = -2, 0 \).- Compute the sign of \( f''(x) \): - \( f''(-2) = 6(-2) + 6 < 0 \) (concave down) - \( f''(0) = 6(0) + 6 > 0 \) (concave up).- The sign diagram shows that the function is concave down on \((-\infty, -1)\) and concave up on \((-1, \infty)\).
07

Sketch the Graph

Use the information from the derivative sign diagrams to sketch the graph:- The function has a local maximum at \( x = -3 \) where it changes from increasing to decreasing.- It has a local minimum at \( x = 1 \) where it changes from decreasing to increasing.- There is an inflection point at \( x = -1 \) where the concavity changes from down to up.- Sketch a curve passing through these points, respecting the intervals of monotonicity and concavity indicated by the sign diagrams.

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 91影视!

Key Concepts

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

Understanding Critical Points
Critical points occur where the first derivative of a function equals zero or is undefined. These points are vital in calculus as they indicate where a function may have a local maximum or minimum. For the function \[ f(x) = x^3 + 3x^2 - 9x + 5 \],we find the first derivative to be \[ f'(x) = 3x^2 + 6x - 9 \].By setting \[ f'(x) \] to zero, we solve for the critical points:
  • Factor the quadratic: \[ 3(x+3)(x-1) = 0 \]
  • Critical points are: \[ x = -3 \] and \[ x = 1 \]
These critical points can signify locations of potential local extrema.
Creating a Derivative Sign Diagram
The first derivative sign diagram helps determine where a function is increasing or decreasing. To build this diagram, we use the critical points we've identified. Here's how:
  • Divide the number line into intervals using the critical points: \((-\infty, -3), (-3, 1), (1, \infty)\).
  • Select test points within each interval: try \( x = -4, 0, \) and \( x = 2 \).
  • Calculate the sign of \( f'(x) \) at these points:
    • \( f'(-4) > 0 \), function is increasing.
    • \( f'(0) < 0 \), function is decreasing.
    • \( f'(2) > 0 \), function is increasing.
This diagram provides a visual representation of where the function ascends and descends, helping predict graph behavior.
Identifying Inflection Points
Inflection points are where the function's concavity changes. To find these, we look at the second derivative. For our function:\[ f''(x) = 6x + 6 \].Setting the second derivative to zero reveals possible inflection points:
  • Solve \[ 6x + 6 = 0 \] to get \[ x = -1 \].
At \( x = -1 \), the function transitions from concave down to concave up, indicating an inflection point on the graph. Confirming these changes is essential in accurately sketching function behavior.
Exploring Concavity
Concavity describes the way a graph curves. A function is concave up where its second derivative is positive, forming a cup shape, and concave down where negative, resembling a cap. For \[ f(x) \]:
  • At \( x = -2 \): \( f''(-2) < 0 \), the function is concave down.
  • At \( x = 0 \): \( f''(0) > 0 \), the function is concave up.
The sign changes at \( x = -1 \) confirm the inflection point. Concavity indicates the graph's direction, showcasing effective interpretation of the graph's natural "bending."
Determining Local Extrema
Local extrema represent the peaks and troughs of a function within a particular interval. Identifying these involves evaluating critical points from the first derivative and what happens around them:
  • At \( x = -3 \), the function shifts from increasing to decreasing, indicating a local maximum.
  • At \( x = 1 \), the function changes from decreasing to increasing, so we have a local minimum.
Incorporating this with concavity insights helps create a vivid picture of these turning points, ensuring a graph is drawn with precision, depicting the function's slope and curvature effectively.

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

Derive the formula \(x=\frac{-0}{2 a}\) for the \(x\) -coordinate of the vertex of parabola \(y=a x^{2}+b x+c\).

Blood flowing through an artery flows faster in the center of the artery and more slowly near the sides (because of friction). The speed of the blood is \(V=c\left(R^{2}-r^{2}\right)\) millimeters \((\mathrm{mm})\) per second, where \(R\) is the radius of the artery, \(r\) is the distance of the blood from the center of the artery, and \(c\) is a constant. Suppose that arteriosclerosis is narrowing the artery at the rate of \(d R / d t=-0.01 \mathrm{~mm}\) per year. Find the rate at which blood flow is being reduced in an artery whose radius is \(R=0.05 \mathrm{~mm}\) with \(c=500\). [Hint: Find \(d V / d t\), considering \(r\) to be a constant. The units of \(d V / d t\) will be \(m m\) per second per year.

Find the linear approximation for each root by choosing the closest value of \(x\) for which the calculation is easy and then using the corresponding value of \(\Delta x\). $$ \sqrt{48} $$

The number \(x\) of handbags that a manufacturer will supply per week and their price \(p\) (in dollars) are related by the equation \(5 x^{3}=20,000+2 p^{2}\). If the price is rising at the rate of \(\$ 2\) per week, find how the supply will change if the current price is \(\$ 100\).

BIOMEDICAL: Bacterial Growth A population of bacteria grows to size \(p(x)=x^{3}-9 x^{2}+24 x+10\) after \(x\) hours (for \(x \geq 0\) ). Graph this population curve (based on, if you wish, a calculator graph), showing the coordinates of the relative extreme points.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation
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.