/*! 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 110 Show that the cubic function $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 110

Show that the cubic function $$ f(x)=a x^{3}+b x^{2}+c x+d \quad(a \neq 0) $$ has no relative extremum if and only if \(b^{2}-3 a c \leq 0\).

Short Answer

Expert verified
A cubic function \(f(x) = ax^3 + bx^2 + cx + d\) with \(a \neq 0\) has no relative extremum if and only if \(b^2 - 3ac \leq 0\). We established this condition by analyzing the discriminant of the first derivative, \(f'(x) = 3ax^2 + 2bx + c\). If \(b^2 - 3ac \leq 0\), the derivative will not have real solutions, indicating no relative extrema in the cubic function.

Step by step solution

01

Finding First Derivative

To find the critical points of the function, we need to find its first derivative. Calculate the first derivative, \(f'(x)\), of the cubic function: $$ f(x) = ax^3 + bx^2 + cx + d $$ Applying the power rule, we get: $$ f'(x) = 3ax^2 + 2bx + c $$
02

Finding Critical Points of the Function

To find the critical points of the function, set \(f'(x)\) equal to zero: $$ 3ax^2 + 2bx + c = 0 $$ We are looking for x-values that satisfy the above equation. This is a quadratic equation in the form of \(Ax^2 + Bx + C = 0\), where \(A = 3a, B = 2b\), and \(C = c\). To determine if the equation has real solutions, we will use the discriminant formula: $$ \Delta = B^2 - 4AC $$ Here, we want to analyze if this quadratic equation has real zeros, which will serve as the critical points for the function's extremum.
03

Evaluating the Discriminant

Substitute the values of \(A, B\), and \(C\) into the discriminant formula: $$ \Delta = (2b)^2 - 4(3a)(c) $$ Simplify the discriminant: $$ \Delta = 4b^2 - 12ac $$ For a quadratic equation, there will be real solutions if \(\Delta > 0\), a repeated solution if \(\Delta = 0\), and no real solutions if \(\Delta < 0\). Since we are trying to prove the condition \(b^2 - 3ac \leq 0\) for the non-existence of relative extrema, let's rewrite the discriminant inequality in the form we need: $$ b^2 - 3ac \geq \frac{1}{3}\Delta $$ In other words, the cubic function will not have a relative extremum if \(b^2 - 3ac \leq 0\) because the derivative will not have real solutions under this condition.
04

Conclusion

A cubic function \(f(x) = ax^3 + bx^2 + cx + d\) with \(a \neq 0\) has no relative extremum if and only if: $$ b^2 - 3ac \leq 0 $$ By analyzing the discriminant of the first derivative, we can deduce this condition.

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.

Critical Points in Calculus
In calculus, critical points are significant because they are the x-values where the slope of the function is zero or undefined; essentially, they're where the graph of the function has a horizontal tangent line or a cusp. These points are monumental in identifying the function's local maxima and minima鈥攃ommonly known as

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

If exactly 200 people sign up for a charter flight, Leisure World Travel Agency charges $$\$300$$/person. However, if more than 200 people sign up for the flight (assume this is the case), then each fare is reduced by $$\$ 1$$ for each additional person. Determine how many passengers will result in a maximum revenue for the travel agency. What is the maximum revenue? What would be the fare per passenger in this case? Hint: Let \(x\) denote the number of passengers above 200 . Show that the revenue function \(R\) is given by \(R(x)=(200+x)(300-x)\).

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ \begin{array}{l} f(x)=\frac{x+1}{x-1} \text { on }[2,4]\\\ \text { 24. } g(t)=\frac{t}{t-1} \text { on }[2,4] \end{array} $$

The percentage of foreign-born residents in the United States from 1910 through 2000 is approximated by the function \(P(t)=0.04363 t^{3}-0.267 t^{2}-1.59 t+14.7 \quad(0 \leq t \leq 9)\) where \(t\) is measured in decades, with \(t=0\) corresponding to \(1910 .\) Show that the percentage of foreign-born residents was lowest in early 1970 . Hint: Use the quadratic formula.

A manufacturer of tennis rackets finds that the total cost \(C(x)\) (in dollars) of manufacturing \(x\) rackets/day is given by \(C(x)=400+4 x+0.0001 x^{2}\). Each racket can be sold at a price of \(p\) dollars, where \(p\) is related to \(x\) by the demand equation \(p=10-0.0004 x\). If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer.

The number of major crimes committed in the city of Bronxville between 2000 and 2007 is approximated by the function $$ N(t)=-0.1 t^{3}+1.5 t^{2}+100 \quad(0 \leq t \leq 7) $$ where \(N(t)\) denotes the number of crimes committed in year \(t(t=0\) corresponds to 2000 ). Enraged by the dramatic increase in the crime rate, the citizens of Bronxville, with the help of the local police, organized "Neighborhood Crime Watch" groups in early 2004 to combat this menace. Show that the growth in the crime rate was maximal in 2005 , giving credence to the claim that the Neighborhood Crime Watch program was working.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.