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.
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.