/*! 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 21 Find the critical numbers of eac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 21

Find the critical numbers of each function. $$ f(x)=-x^{4}+4 x^{3}-4 x^{2}+1 $$

Short Answer

Expert verified
The critical numbers are \( x = 0 \), \( x = 1 \), and \( x = 2 \).

Step by step solution

01

Find the derivative

To find the critical numbers, first, we need to find the derivative of the function. The function given is \[ f(x) = -x^4 + 4x^3 - 4x^2 + 1. \]The derivative, \( f'(x) \), is calculated as follows:- The derivative of \( -x^4 \) is \( -4x^3 \).- The derivative of \( 4x^3 \) is \( 12x^2 \).- The derivative of \( -4x^2 \) is \( -8x \).- The derivative of the constant \( 1 \) is \( 0 \).Putting it all together:\[ f'(x) = -4x^3 + 12x^2 - 8x. \]
02

Set the derivative to zero

To find the critical numbers, set the derivative equal to zero:\[ -4x^3 + 12x^2 - 8x = 0. \]
03

Factor the derivative equation

Factor the equation solved in Step 2:First, factor out the greatest common factor, which is \( -4x \):\[ -4x(x^2 - 3x + 2) = 0. \]Next, factor the quadratic equation:\[ x^2 - 3x + 2 = (x - 1)(x - 2). \]Thus, the equation becomes:\[ -4x(x - 1)(x - 2) = 0. \]
04

Solve the factored equation

Set each factor equal to zero and solve for \( x \):1. \( -4x = 0 \) gives \( x = 0 \).2. \( x - 1 = 0 \) gives \( x = 1 \).3. \( x - 2 = 0 \) gives \( x = 2 \).Thus, the critical numbers are \( x = 0 \), \( x = 1 \), and \( x = 2 \).

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.

Derivative Calculation
Finding the derivative is an essential step in solving many calculus problems. A derivative gives us the rate at which a function is changing at any given point. It involves differentiating each term of the function separately and then combining these derivative terms. For our function, \( f(x) = -x^4 + 4x^3 - 4x^2 + 1 \), the derivative calculation proceeds as follows:
  • Differentiate each term separately.
  • The derivative of \( -x^4 \) is \( -4x^3 \). Simply multiply the power by the coefficient and reduce the power by one.
  • The derivative of \( 4x^3 \) is \( 12x^2 \).
  • The derivative of \( -4x^2 \) is \( -8x \).
  • Finally, the derivative of a constant, such as \( 1 \), is \( 0 \).
Putting these all together, we get the derivative \( f'(x) = -4x^3 + 12x^2 - 8x \). Calculating derivatives, learning these basic rules, and doing plenty of practice can make this step much easier.
Polynomial Function
A polynomial function is made up of terms that are non-negative integer powers of a variable. In our example, the function \( f(x) = -x^4 + 4x^3 - 4x^2 + 1 \) is a typical polynomial. Key characteristics of polynomial functions include:
  • They can have constants, variables, and exponents.
  • The exponents must be whole numbers.
  • They should not have division by the variable, negative exponents, or variables under a root sign.
Polynomial functions are continuous and smooth over their domain, which is usually all real numbers. They enable us to model a large variety of natural and physical phenomena, making them very versatile in both basic and advanced mathematics.
Factoring Equations
Factoring is a crucial skill in solving polynomial equations, especially when finding critical numbers where derivatives equal zero. Factoring involves writing polynomial expressions as a product of their simpler expressions. Let's consider how we factored the derivative equation:
  • Initially, factor out the greatest common factor. In this case, \( -4x \) is common in all terms of \( -4x^3 + 12x^2 - 8x \), so we write it as \( -4x(x^2 - 3x + 2) \).
  • Next, factor the quadratic \( x^2 - 3x + 2 \) further into \( (x - 1)(x - 2) \). We use methods such as testing factors of the constant term and learning to recognize patterns.
The factored form, \( -4x(x - 1)(x - 2) \), simplifies the calculation of solutions since setting each factor to zero gives us the critical points efficiently and accurately.
Calculus Problem Solving
Solving calculus problems involves a structured approach. The goal is often to find values like critical numbers which inform us about the characteristics of a function graph. For the problem at hand, follow these steps:
  • Find the derivative of the function to understand how it changes.
  • Set this derivative to zero to find critical points, which are x-values where the function's slope is zero, marking peaks or troughs.
  • Factor the derivative equation to simplify finding these solutions, using greatest common factors and further polynomial factorization.
  • Solve for the zero-points from the factored terms. Each solution can indicate a point of interest in the graph's behavior.
In our exercise, these steps allow us to pinpoint critical numbers: \( x = 0 \), \( x = 1 \), and \( x = 2 \). These points are crucial for understanding where the graph changes its direction and help provide insights into its overall shape and nature.

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

The value \(V(t)\) of a case of investment-grade wine after \(t\) years is $$ V(t)=2000+80 \sqrt{t}-10 t $$ dollars (for \(0 \leq t \leq 25\) ). For each number of years, find \(d V\) and use it to estimate the value one year later. $$ \text { 9 years } $$

The number \(x\) of MP3 music players that a store will sell and their price \(p\) (in dollars) are related by the equation \(2 x^{2}=15,000-p^{2} .\) Find \(d x / d p\) at \(p=100\) and interpret your answer. [Hint: You will have to find the value of \(x\) by substituting the given value of \(p\) into the original equation.

In general, implicit differentiation gives an expression for the derivative that involves both \(x\) and \(y\). Under what conditions will the expression involve only \(x ?\)

\(33-62 .\) Sketch the graph of each rational function after making a sign diagram for the derivative and finding all relative extreme points and asymptotes. $$ f(x)=\frac{16}{(x+2)^{3}} $$

Steel ball bearings are manufactured with a diameter error of no more than \(\pm 0.1 \%\). What is the relative error in the volume?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.