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Chapter 3: Problem 81

Why is a third-degree polynomial function with a negative leading coefficient not appropriate for modeling nonnegative real-world phenomena over a long period of time?

Short Answer

Expert verified
A third-degree polynomial function with a negative leading coefficient is inappropriate for modeling nonnegative real-world phenomena over a long period of time because it tends towards negative infinity with increasing 'x' values, which is unrealistic for such phenomena that cannot be negative.

Step by step solution

01

Understanding Third-Degree Polynomial Functions

Third-degree polynomial or cubic functions are represented by \(f(x) = ax^{3} + bx^{2} + cx + d\), where 'a' is the leading coefficient. When 'a' is positive, as 'x' tends towards positive or negative infinity, 'f(x)' also tends towards positive or negative infinity respectively. If 'a' is negative, the behavior is reversed.
02

End Behavior According to the Coefficient

For a third-degree polynomial with a negative leading coefficient, as 'x' values move towards positive infinity, 'f(x)' values will tend towards negative infinity. This indicates that the function will decrease without bound as time progresses.
03

Cubic Function for Positive Phenomena

If the function is used to model nonnegative real-world phenomena (for example, population, revenue, etc.), it's not realistic over a long period of time as these phenomena cannot be negative. Therefore, a cubic function with a negative leading coefficient is inappropriate for such scenarios over a long period.

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