Select the correct answer.
| x | f(x) |
| --- | ----- |
| 3.0 | 4.0 |
| 3.5 | -0.2 |
| 4.0 | -0.8 |
| 4.5 | 0.1 |
| 5.0 | 0.6 |
| 5.5 | 0.7 |
For the given table of values for a polynomial function, where must the zeros of the function lie?
A. between 3.0 and 3.5 and between 4.0 and 4.5
B. between 3.5 and 4.0 and between 4.0 and 4.5
C. between 3.5 and 4.0 and between 5.0 and 5.5
D. between 4.0 and 4.5 and between 4.5 and 5.0
Answer (2)
The zeros of the polynomial function must lie in the intervals where the function changes sign. Based on the given table, the function changes sign between x = 3.0 and x = 3.5 , and between x = 4.0 and x = 4.5 . Therefore, the zeros must lie between 3.0 and 3.5 and between 4.0 and 4.5. A
Explanation
Understanding the Problem We are given a table of values for a polynomial function f ( x ) and asked to determine the intervals in which the zeros of the function must lie. A zero of a function is a value x such that f ( x ) = 0 . According to the Intermediate Value Theorem, if a continuous function changes sign between two points, there must be a zero between those points.
Identifying Sign Changes We need to check for sign changes in the values of f ( x ) in the table.
From x = 3.0 to x = 3.5 , f ( x ) changes from 4.0 to − 0.2 . Since there is a sign change, there must be a zero between 3.0 and 3.5 .
From x = 3.5 to x = 4.0 , f ( x ) changes from − 0.2 to − 0.8 . There is no sign change here.
From x = 4.0 to x = 4.5 , f ( x ) changes from − 0.8 to 0.1 . Since there is a sign change, there must be a zero between 4.0 and 4.5 .
From x = 4.5 to x = 5.0 , f ( x ) changes from 0.1 to 0.6 . There is no sign change here.
From x = 5.0 to x = 5.5 , f ( x ) changes from 0.6 to 0.7 . There is no sign change here.
Determining Intervals with Zeros The intervals where the sign of f ( x ) changes are between 3.0 and 3.5 , and between 4.0 and 4.5 . Therefore, the zeros of the function must lie between 3.0 and 3.5 and between 4.0 and 4.5 .
Selecting the Correct Answer Comparing the identified intervals with the options provided, we see that option A matches our result: between 3.0 and 3.5 and between 4.0 and 4.5.
Examples
The Intermediate Value Theorem, which helps us find the zeros of a function, is used in many real-world applications. For example, if you are designing a bridge, you need to ensure that the stress on the bridge is zero at certain points. Similarly, in economics, you might want to find the equilibrium point where the supply and demand curves intersect, which is where the difference between supply and demand is zero. These problems can be solved by finding the zeros of a function.
The zeros of the polynomial function lie in the intervals between 3.0 and 3.5, and between 4.0 and 4.5. This is determined by observing sign changes in the function values. Thus, the correct answer is A.
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