Answer :
Let's analyze the given table of values:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -3 & -16 \\ \hline -2 & -1 \\ \hline -1 & 2 \\ \hline 0 & -1 \\ \hline 1 & -4 \\ \hline 2 & -1 \\ \hline \end{tabular} \][/tex]
### Finding Local Maximum
A local maximum occurs where a function changes from increasing to decreasing.
1. Compare [tex]$f(x)$[/tex] values at [tex]$x=-2, -1, 0$[/tex]:
- [tex]$f(-2) = -1$[/tex], [tex]$f(-1) = 2$[/tex], [tex]$f(0) = -1$[/tex]
- Here, [tex]$f(x)$[/tex] changes from [tex]$-1$[/tex] (at [tex]$x = -2$[/tex]) to [tex]$2$[/tex] (at [tex]$x = -1$[/tex]) then to [tex]$-1$[/tex] (at [tex]$x = 0$[/tex]).
2. Notice that at [tex]$x = -1$[/tex], the function increases from [tex]$x = -2$[/tex] and then decreases towards [tex]$x = 0$[/tex]. Therefore, this is a local maximum.
3. The interval of this occurrence is [tex]$(-2, 0)$[/tex].
### Finding Local Minimum
A local minimum occurs where a function changes from decreasing to increasing.
1. Compare [tex]$f(x)$[/tex] values at [tex]$x=0, 1, 2$[/tex]:
- [tex]$f(0) = -1$[/tex], [tex]$f(1) = -4$[/tex], [tex]$f(2) = -1$[/tex]
- Here, [tex]$f(x)$[/tex] changes from [tex]$-1$[/tex] (at [tex]$x = 0$[/tex]) to [tex]$-4$[/tex] (at [tex]$x = 1$[/tex]) then to [tex]$-1$[/tex] (at [tex]$x = 2$[/tex]).
2. Notice that at [tex]$x = 1$[/tex], the function decreases from [tex]$x = 0$[/tex] and then increases towards [tex]$x = 2$[/tex]. Therefore, this is a local minimum.
3. The interval of this occurrence is [tex]$(0, 2)$[/tex].
### Conclusion
Thus, analyzing the table values and applying the conditions for local maxima and minima, we get:
- A local maximum occurs over the interval [tex]\((-2, 0)\)[/tex].
- A local minimum occurs over the interval [tex]\((0, 2)\)[/tex].
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -3 & -16 \\ \hline -2 & -1 \\ \hline -1 & 2 \\ \hline 0 & -1 \\ \hline 1 & -4 \\ \hline 2 & -1 \\ \hline \end{tabular} \][/tex]
### Finding Local Maximum
A local maximum occurs where a function changes from increasing to decreasing.
1. Compare [tex]$f(x)$[/tex] values at [tex]$x=-2, -1, 0$[/tex]:
- [tex]$f(-2) = -1$[/tex], [tex]$f(-1) = 2$[/tex], [tex]$f(0) = -1$[/tex]
- Here, [tex]$f(x)$[/tex] changes from [tex]$-1$[/tex] (at [tex]$x = -2$[/tex]) to [tex]$2$[/tex] (at [tex]$x = -1$[/tex]) then to [tex]$-1$[/tex] (at [tex]$x = 0$[/tex]).
2. Notice that at [tex]$x = -1$[/tex], the function increases from [tex]$x = -2$[/tex] and then decreases towards [tex]$x = 0$[/tex]. Therefore, this is a local maximum.
3. The interval of this occurrence is [tex]$(-2, 0)$[/tex].
### Finding Local Minimum
A local minimum occurs where a function changes from decreasing to increasing.
1. Compare [tex]$f(x)$[/tex] values at [tex]$x=0, 1, 2$[/tex]:
- [tex]$f(0) = -1$[/tex], [tex]$f(1) = -4$[/tex], [tex]$f(2) = -1$[/tex]
- Here, [tex]$f(x)$[/tex] changes from [tex]$-1$[/tex] (at [tex]$x = 0$[/tex]) to [tex]$-4$[/tex] (at [tex]$x = 1$[/tex]) then to [tex]$-1$[/tex] (at [tex]$x = 2$[/tex]).
2. Notice that at [tex]$x = 1$[/tex], the function decreases from [tex]$x = 0$[/tex] and then increases towards [tex]$x = 2$[/tex]. Therefore, this is a local minimum.
3. The interval of this occurrence is [tex]$(0, 2)$[/tex].
### Conclusion
Thus, analyzing the table values and applying the conditions for local maxima and minima, we get:
- A local maximum occurs over the interval [tex]\((-2, 0)\)[/tex].
- A local minimum occurs over the interval [tex]\((0, 2)\)[/tex].
