30 Dec 2021
Problem 66b
Page 270
Section 4.2: Maximum and Minimum Values
Chapter 4: Applications of Differentiation
Textbook ExpertVerified Tutor
30 Dec 2021
Given information
Given that a cubic function is a polynomial of degree 3 ; it has the form 
, where 
Step-by-step explanation
Step 1.
A function has a local extrema only when its derivative changes sign. Note that : Derivative is zero is necessary but not sufficient condition.
The derivative any cubic polynomial is quadratic polynomial.
There are three types of quadratic polynomials:
Type 1: Quadratic polynomial with no zero.
Which means the derivative will not change its sign and hence no local extrema.
Type 2: Quadratic polynomial has one zero.
But when a quadratic polynomial has only one zero its graph just touches the 
-axis, it does not change its sign in this case too.
-axis, it does not change its sign in this case too.Type 3: Quadratic polynomial has two zeros.
The quadratic polynomial which falls in this category changes its sign twice.
Hence there will be two local extrema.
Therefore we conclude that:
a cubic polynomial can either have two local extrema or no local extrema.
Single Variable Calculus: Early Transcendentals
4th Edition, 2018
Stewart
ISBN: 9781337687805
Solutions
Chapter
Section
- Problem 1
- Problem 2a
- Problem 2b
- Problem 3
- Problem 4
- Problem 5
- Problem 6
- Problem 7
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- Problem 11a
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- Problem 12a
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- Problem 60a
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- Problem 61
- Problem 62
- Problem 63
- Problem 64a
- Problem 64b
- Problem 65a
- Problem 65b
- Problem 65c
- Problem 66a
- Problem 66b