/*! 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 Give an example of a cubic funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 7

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f\) is always decreasing. \(f(3)=0\) and \(f(0)=2\).

Short Answer

Expert verified
A possible cubic function that fulfills the given conditions is \(f(x) = -x^3 + 2\).

Step by step solution

01

Formulate the General Form of Cubic Function

The general form of a cubic function is \(f(x) = ax^3 + bx^2 + cx + d\). Based on the problem specifications, our coefficient \(a\) for the term \(x^3\) should be negative because the function is always decreasing. Now we have \(f(x) = -ax^3 + bx^2 + cx + d\).
02

Substitute the Point (3,0) into the Equation

We're given that the function equals 0 when \(x = 3\). We can substitute these values into our function and set the equation equal to zero: \(0 = -a(3)^3 + b(3)^2 + c(3) + d\). Simplifying, we get \(0 = -27a + 9b + 3c + d\). This is our first equation for the system of equations.
03

Substitute the Point (0,2) into the Equation

The function equals 2 when \(x = 0\). Substitute these into the function: \(2 = -a(0)^3 + b(0)^2 + c(0) + d\), or simply, \(d = 2\).
04

Formulate the Cubic Function

Since we only need to provide one example of a cubic function that meets these conditions, we can let \(a = 1\), \(b = 0\) and \(c = 0\) for simplicity. This simplifies our equation from step 2: \(0 = -27(1) + 9(0) + 3(0) + 2\). This gives us \(a = 1\). With \(a = 1\), \(b = 0\), \(c = 0\), and \(d = 2\), our cubic function becomes: \(f(x) = -x^3 + 2\).
05

Validate the Function

It's important to validate that the function \(f(x) = -x^3 + 2\) meets all the original conditions specified in the problem. Indeed, \(f(x)\) is always decreasing, \(f(3) = 0\), and \(f(0) = 2\).

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.

Decreasing Function
A function is considered "decreasing" when its outputs consistently get smaller as the inputs (or "x" values) increase. This means that if you pick any two different points on the graph of the function, the function value (or "y" value) is smaller for the larger "x" value.
A function decreases because its derivative is negative over its domain. In simpler terms, the slope or gradient of the function is downward or negative at every point.
For our cubic function example, the derivative of the function is always negative, ensuring that the function is indeed always decreasing. This is important to check when creating or analyzing functions under these conditions.
Function Evaluation
Function evaluation is the process of finding the output or function value given a particular input ("x" value). You simply plug the "x" value into the function's equation to find the corresponding "y" value.
In the exercise at hand, the function was evaluated at specific points:
  • At point \( x = 3 \), the function is evaluated to verify that \( f(3) = 0 \).
  • At point \( x = 0 \), the function is evaluated to confirm that \( f(0) = 2 \).
This process ensures that conditions given in the problem are met, such as specific zeros of the function or other intercepts.
System of Equations
A system of equations is a set of two or more equations that you deal with simultaneously. Solutions to the system need to satisfy all equations involved. Often, systems of equations are used to find values of unknown variables.
In this example problem, we derived a system of equations from the conditions specified:
  • From \( f(3) = 0 \), we obtained: \( 0 = -27a + 9b + 3c + d \).
  • From \( f(0) = 2 \), we derived: \( d = 2 \).
By solving these equations, specific values for coefficients \( a \), \( b \), \( c \), and \( d \) were determined, helping construct the desired cubic function.
Cubic Polynomial
A cubic polynomial is a polynomial of degree 3, which means it includes a term with \( x^3 \) as the highest power of \( x \). The general form of a cubic function is given by \( f(x) = ax^3 + bx^2 + cx + d \).
This type of function can have varied shapes, often depending on the coefficients of the terms. The leading coefficient, which is the coefficient of \( x^3 \), is crucial in determining the shape and behavior of the graph.
  • If the leading coefficient is positive, the ends of the curve go to opposite directions; pointing up as \( x \) heads towards infinity, and down as \( x \) goes to negative infinity.
  • If it is negative, as with \( f(x) = -x^3 + 2 \), the end behavior is reversed, and the curve moves down as \( x \) increases.
Understanding the structure and behavior of cubic polynomials is important for constructing and analyzing these types of functions.

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) Suppose \(P(x)\) is a polynomial of degree \(6 .\) Which of the statements that follow must necessarily be true? If a statement is not necessarily true, provide a counterexample (an example for which the statement is false). i. \(P(x)\) has at least one zero. ii. \(P(x)\) has no more than five zeros. iii. The graph of \(P(x)\) has at least one turning point. iv. The graph of \(P(x)\) has at most five turning points. (b) Suppose \(P(x)\) is a polynomial of degree 6 with its natural domain \((-\infty, \infty)\). If \(P^{\prime}(2)=0\) and \(P^{\prime \prime}(2)=-1\), then which one of the following statements is true? Explain your answer. i. \(P\) has a local minimum at \(x=2\) but this local minimum is not an absolute minimum. ii. \(P\) has a local minimum at \(x=2\) and this local minimum may be an absolute minimum. iii. \(P\) has a local maximum at \(x=2\) but this local maximum is not an absolute maximum. iv. \(P\) has a local maximum at \(x=2\) and this local maximum may be an absolute maximum.

Suppose a distance function is given by \(d(t)=1 / t\) for \(0.5 \leq t \leq 20\). (a) What is the average velocity over the interval from \(t=1\) to \(t=5\) ? (b) Is there a time at which the instantaneous velocity is the same as the average velocity over the interval from \(t=1\) to \(t=5 ?\) If so, find that time. (c) On the same set of axes, illustrate your answers to parts (a) and (b).

Graph the function given, labeling all \(x\) -intercepts, \(y\) intercepts, and the \(x\) - and \(y\) -coordinates of any local maximum and minimum points. $$ f(x)=x(x-2)^{2} $$

Construct a polynomial \(P(x)\) with the specified characteristics. Determine whether or not the answer to the problem is unique. Explain and/or illustrate your answer. A third degree polynomial with zeros at \(x=1\) and \(x=2\), a turning point at \(x=1\), and a \(y\) -intercept of \(\sqrt{e}\).

Graph the function given, labeling all \(x\) -intercepts, \(y\) intercepts, and the \(x\) - and \(y\) -coordinates of any local maximum and minimum points. $$ f(x)=x^{2}(x-2) $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.