To determine where the zeros of the function lie, we need to identify the intervals where the function changes sign. A sign change between two consecutive values of [tex]\( f(x) \)[/tex] indicates that a zero exists between the corresponding [tex]\( x \)[/tex]-values.
Given the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
3.0 & 4.0 \\
\hline
3.5 & -0.2 \\
\hline
4.0 & -0.8 \\
\hline
4.5 & 0.1 \\
\hline
5.0 & 0.6 \\
\hline
5.5 & 0.7 \\
\hline
\end{array}
\][/tex]
We need to check each consecutive pair to identify where the function changes from positive to negative or from negative to positive.
1. Between [tex]\( x = 3.0 \)[/tex] and [tex]\( x = 3.5 \)[/tex]:
- [tex]\( f(3.0) = 4.0 \)[/tex] (positive)
- [tex]\( f(3.5) = -0.2 \)[/tex] (negative)
- There is a sign change from positive to negative.
- Therefore, a zero lies between [tex]\( x = 3.0 \)[/tex] and [tex]\( x = 3.5 \)[/tex].
2. Between [tex]\( x = 3.5 \)[/tex] and [tex]\( x = 4.0 \)[/tex]:
- [tex]\( f(3.5) = -0.2 \)[/tex] (negative)
- [tex]\( f(4.0) = -0.8 \)[/tex] (negative)
- No sign change.
3. Between [tex]\( x = 4.0 \)[/tex] and [tex]\( x = 4.5 \)[/tex]:
- [tex]\( f(4.0) = -0.8 \)[/tex] (negative)
- [tex]\( f(4.5) = 0.1 \)[/tex] (positive)
- There is a sign change from negative to positive.
- Therefore, a zero lies between [tex]\( x = 4.0 \)[/tex] and [tex]\( x = 4.5 \)[/tex].
4. Between [tex]\( x = 4.5 \)[/tex] and [tex]\( x = 5.0 \)[/tex]:
- [tex]\( f(4.5) = 0.1 \)[/tex] (positive)
- [tex]\( f(5.0) = 0.6 \)[/tex] (positive)
- No sign change.
5. Between [tex]\( x = 5.0 \)[/tex] and [tex]\( x = 5.5 \)[/tex]:
- [tex]\( f(5.0) = 0.6 \)[/tex] (positive)
- [tex]\( f(5.5) = 0.7 \)[/tex] (positive)
- No sign change.
Based on this analysis, the zeros of the function must lie in the intervals:
- Between [tex]\( 3.0 \)[/tex] and [tex]\( 3.5 \)[/tex]
- Between [tex]\( 4.0 \)[/tex] and [tex]\( 4.5 \)[/tex]
Therefore, the correct answer is:
A. between 3.0 and 3.5 and between 4.0 and 4.5.
