   Chapter 4.3, Problem 73E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of 3 at x = −2 and a local minimum value of 0 at x = 1.

To determine

To find: A cubic function with the given criteria.

Explanation

Given:

The function is f(x)=ax3+bx2+cx+d and it has local maximum value f(2)=3 and local minimum value f(1)=0.

Calculation:

The given function is, f(x)=ax3+bx2+cx+d (1)

Substitute f(2)=3 in equation (1),

f(2)=3a(2)3+b(2)2+c(2)+d=3a(8)+b(4)+c(2)+d3=0

8a+4b2c+d=3 (2)

Substitute f(1)=0 in equation (1),

f(1)=0a(1)3+b(1)2+c(1)+d=0

a+b+c+d=0 (3)

Obtain the derivative of the function f(x).

f(x)=ddx(ax3+bx2+cx+d)=a(3x2)+b(2x)+c(1)+0

f(x)=3ax2+2bx+c (4)

Since the local maximum is occurred at the point 2, by the derivative test f(2)=0 and substitute in equation (4),

f(2)=03a(2)2+2b(

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