   Chapter 3.1, Problem 72E

Chapter
Section
Textbook Problem

# A cubic function is a polynomial of degree 3; that is, it has the form f ( x ) = a x 3 + b x 2 + c x + d , where a ≠ 0 .(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.(b) How many local extreme values can a cubic function have?

To determine

Part (a):

To show:

That a cubic function can have two, one, or no critical number and give examples and sketches to illustrate the three possibilities.

Explanation

1) Concept:

A critical number of a function f   is a number c in the domain of f  such that either  f'c=0 or f'c does not exist.

2) Given:

fx=ax3+bx2+cx+d where a0

3) Calculation:

To find the critical number, differentiate fx using the power rule of derivative.

f'x=3ax2+2bx1+c

This is a quadratic polynomial of degree two.

Therefore, the polynomial has at the most two real roots.

So fx has either 2 , 1 or 0  critical number.

Case (I): 2 critical numbers

Consider fx=2x3-6x

Differentiate fx by using the power rule of derivative.

f'x=6x2-6

To find critical numbers, set f'x=0, and solve for x.

6x2-6=0

Therefore, x=±1 are the two critical numbers since f'x exists for all real numbers.

Graph of fx with 2 critical numbers

Case (II): One critical number

Consider fx=23x3

Differentiate fx with respect to x by using the power rule of derivative

To determine

Part (b):

To find:

Number of local extreme values for a cubic function

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Evaluate the integral. 22f(x)dxwheref(x)={2if2x04x2if0x2

Single Variable Calculus: Early Transcendentals, Volume I

#### Find f(x) if f(x)=1x2 and f(2) = 0. a) 1x+12 b) 1x12 c) 3x3+38 d) 3x338

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

#### What are the five steps of the scientific method?

Research Methods for the Behavioral Sciences (MindTap Course List) 