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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A cubic function is a polynomial of degree 3; that is, it has the form f(x) = ax3 + bx2 + cx + 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?

(a)

To determine

To show: A cubic function f(x)=ax3+bx2+cx+d , where a0 can have two, one, or no critical number(s); sketch the graph of those three possibilities.

Explanation

The general equation of the cubic polynomial is f(x)=ax3+bx2+cx+d , where a0 .

Obtain the first derivative of f(x) .

f(x)=ddx(ax3+bx2+cx+d)=3ax2+2bx+c

Since the degree of f'(x) is two, f(x)=3ax2+2bx+c is a quadratic function with two, one or zero real roots.

That is, f(x) can have 2, 1 or 0 critical numbers.

The three different possibilities along with the graphs are explained below:

Case 1: Two critical numbers

Consider the cubic function is f(x)=3x39x .

Obtain the derivative of f(x) .

f(x)=ddx(3x39x)=3(3x2)9(1)=9x29=9(x21)

Set f(x)=0 to obtain the critical numbers.

9(x21)=0x21=0x2=1x=±1

The critical numbers are x=1,x=1 .

The sketch of f(x)=3x39x is shown below in Figure 1.

From Figure 1, it is observed that the graph changes its direction at the critical numbers

(b)

To determine

To find: The possible number of local extreme values of a cubic function.

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