Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.