Expand your knowledge base with the help of IDNLearn.com's extensive answer archive. Join our interactive Q&A platform to receive prompt and accurate responses from experienced professionals in various fields.
Sagot :
To find the local minimum of the function [tex]\( f(x) \)[/tex] using the given table of values, we need to identify points where the function value is less than that of its neighboring points. A local minimum is a point in the function where the function value at that point is less than the function values at neighboring points.
Here is the table again for reference:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -4 & 105 \\ \hline -3 & 0 \\ \hline -2 & -15 \\ \hline -1 & 0 \\ \hline 0 & 9 \\ \hline 1 & 0 \\ \hline 2 & -15 \\ \hline 3 & 0 \\ \hline 4 & 105 \\ \hline 5 & 384 \\ \hline \end{tabular} \][/tex]
Consider each point and its immediate neighbors:
1. For [tex]\( x = -4 \)[/tex], we don't have neighbors on both sides, so we can't determine if it's a local minimum.
2. For [tex]\( x = -3 \)[/tex], the neighbors are [tex]\( f(-4) = 105 \)[/tex] and [tex]\( f(-2) = -15 \)[/tex]. Here, [tex]\( f(-2) = -15 < f(-3) = 0 \)[/tex], so [tex]\( x = -3 \)[/tex] is not a local minimum.
3. For [tex]\( x = -2 \)[/tex], the neighbors are [tex]\( f(-3) = 0 \)[/tex] and [tex]\( f(-1) = 0 \)[/tex]. Here, [tex]\( f(-2) = -15 \)[/tex] is less than both [tex]\( f(-3) \)[/tex] and [tex]\( f(-1) \)[/tex]. This indicates that [tex]\( x = -2 \)[/tex] is a local minimum.
4. For [tex]\( x = -1 \)[/tex], the neighbors are [tex]\( f(-2) = -15 \)[/tex] and [tex]\( f(0) = 9 \)[/tex]. Here, [tex]\( f(-2) = -15 < f(-1) = 0 \)[/tex], so [tex]\( x = -1 \)[/tex] is not a local minimum.
5. For [tex]\( x = 0 \)[/tex], the neighbors are [tex]\( f(-1) = 0 \)[/tex] and [tex]\( f(1) = 0 \)[/tex]. Here, [tex]\( f(0) = 9 \)[/tex] is greater than both [tex]\( f(-1) \)[/tex] and [tex]\( f(1) \)[/tex], so [tex]\( x = 0 \)[/tex] is not a local minimum.
6. For [tex]\( x = 1 \)[/tex], the neighbors are [tex]\( f(0) = 9 \)[/tex] and [tex]\( f(2) = -15 \)[/tex]. Here, [tex]\( f(2) = -15 < f(1) = 0 \)[/tex], so [tex]\( x = 1 \)[/tex] is not a local minimum.
7. For [tex]\( x = 2 \)[/tex], the neighbors are [tex]\( f(1) = 0 \)[/tex] and [tex]\( f(3) = 0 \)[/tex]. Here, [tex]\( f(2) = -15 \)[/tex] is less than both [tex]\( f(1) \)[/tex] and [tex]\( f(3) \)[/tex]. This indicates that [tex]\( x = 2 \)[/tex] is a local minimum.
8. For [tex]\( x = 3 \)[/tex], the neighbors are [tex]\( f(2) = -15 \)[/tex] and [tex]\( f(4) = 105 \)[/tex]. Here, [tex]\( f(2) = -15 < f(3) = 0 \)[/tex], so [tex]\( x = 3 \)[/tex] is not a local minimum.
9. For [tex]\( x = 4 \)[/tex], the neighbors are [tex]\( f(3) = 0 \)[/tex] and [tex]\( f(5) = 384 \)[/tex]. Here, [tex]\( f(3) = 0 < f(4) = 105 \)[/tex], so [tex]\( x = 4 \)[/tex] is not a local minimum.
10. For [tex]\( x = 5 \)[/tex], we don't have neighbors on both sides, so we can't determine if it's a local minimum.
Thus, we identify [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex] as local minima. Given the choices:
- [tex]\((0, 9)\)[/tex]
- [tex]\((4, 105)\)[/tex]
- [tex]\((-1, 0)\)[/tex]
- [tex]\((2, -15)\)[/tex]
We notice that [tex]\((2, -15)\)[/tex] is among the options and is indeed one of the local minima identified.
Therefore, the ordered pair that is a local minimum of the function [tex]\( f(x) \)[/tex] is [tex]\((2, -15)\)[/tex].
Here is the table again for reference:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -4 & 105 \\ \hline -3 & 0 \\ \hline -2 & -15 \\ \hline -1 & 0 \\ \hline 0 & 9 \\ \hline 1 & 0 \\ \hline 2 & -15 \\ \hline 3 & 0 \\ \hline 4 & 105 \\ \hline 5 & 384 \\ \hline \end{tabular} \][/tex]
Consider each point and its immediate neighbors:
1. For [tex]\( x = -4 \)[/tex], we don't have neighbors on both sides, so we can't determine if it's a local minimum.
2. For [tex]\( x = -3 \)[/tex], the neighbors are [tex]\( f(-4) = 105 \)[/tex] and [tex]\( f(-2) = -15 \)[/tex]. Here, [tex]\( f(-2) = -15 < f(-3) = 0 \)[/tex], so [tex]\( x = -3 \)[/tex] is not a local minimum.
3. For [tex]\( x = -2 \)[/tex], the neighbors are [tex]\( f(-3) = 0 \)[/tex] and [tex]\( f(-1) = 0 \)[/tex]. Here, [tex]\( f(-2) = -15 \)[/tex] is less than both [tex]\( f(-3) \)[/tex] and [tex]\( f(-1) \)[/tex]. This indicates that [tex]\( x = -2 \)[/tex] is a local minimum.
4. For [tex]\( x = -1 \)[/tex], the neighbors are [tex]\( f(-2) = -15 \)[/tex] and [tex]\( f(0) = 9 \)[/tex]. Here, [tex]\( f(-2) = -15 < f(-1) = 0 \)[/tex], so [tex]\( x = -1 \)[/tex] is not a local minimum.
5. For [tex]\( x = 0 \)[/tex], the neighbors are [tex]\( f(-1) = 0 \)[/tex] and [tex]\( f(1) = 0 \)[/tex]. Here, [tex]\( f(0) = 9 \)[/tex] is greater than both [tex]\( f(-1) \)[/tex] and [tex]\( f(1) \)[/tex], so [tex]\( x = 0 \)[/tex] is not a local minimum.
6. For [tex]\( x = 1 \)[/tex], the neighbors are [tex]\( f(0) = 9 \)[/tex] and [tex]\( f(2) = -15 \)[/tex]. Here, [tex]\( f(2) = -15 < f(1) = 0 \)[/tex], so [tex]\( x = 1 \)[/tex] is not a local minimum.
7. For [tex]\( x = 2 \)[/tex], the neighbors are [tex]\( f(1) = 0 \)[/tex] and [tex]\( f(3) = 0 \)[/tex]. Here, [tex]\( f(2) = -15 \)[/tex] is less than both [tex]\( f(1) \)[/tex] and [tex]\( f(3) \)[/tex]. This indicates that [tex]\( x = 2 \)[/tex] is a local minimum.
8. For [tex]\( x = 3 \)[/tex], the neighbors are [tex]\( f(2) = -15 \)[/tex] and [tex]\( f(4) = 105 \)[/tex]. Here, [tex]\( f(2) = -15 < f(3) = 0 \)[/tex], so [tex]\( x = 3 \)[/tex] is not a local minimum.
9. For [tex]\( x = 4 \)[/tex], the neighbors are [tex]\( f(3) = 0 \)[/tex] and [tex]\( f(5) = 384 \)[/tex]. Here, [tex]\( f(3) = 0 < f(4) = 105 \)[/tex], so [tex]\( x = 4 \)[/tex] is not a local minimum.
10. For [tex]\( x = 5 \)[/tex], we don't have neighbors on both sides, so we can't determine if it's a local minimum.
Thus, we identify [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex] as local minima. Given the choices:
- [tex]\((0, 9)\)[/tex]
- [tex]\((4, 105)\)[/tex]
- [tex]\((-1, 0)\)[/tex]
- [tex]\((2, -15)\)[/tex]
We notice that [tex]\((2, -15)\)[/tex] is among the options and is indeed one of the local minima identified.
Therefore, the ordered pair that is a local minimum of the function [tex]\( f(x) \)[/tex] is [tex]\((2, -15)\)[/tex].
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.
