Answer :
To determine the intervals that contain local maxima and minima based on the given [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] values, we need to analyze the behavior of [tex]\( f(x) \)[/tex] at each point.
Given the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & -54 & -20 & -4 & 0 & -2 & -4 & 0 & 16 & 50 \\ \hline \end{array} \][/tex]
We will look for the positions where [tex]\( f(x) \)[/tex] is greater than its neighbors (indicating a local maximum) or less than its neighbors (indicating a local minimum).
1. Local Maximum:
A local maximum occurs where the value of [tex]\( f(x) \)[/tex] at a point is greater than the values at neighboring points.
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -20 \)[/tex]:
- [tex]\( f(-4) = -54 \)[/tex] (left neighbor)
- [tex]\( f(-2) = -4 \)[/tex] (right neighbor)
- [tex]\( -20 \)[/tex] is not greater than both [tex]\(-54\)[/tex] and [tex]\(-4\)[/tex].
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -4 \)[/tex]:
- [tex]\( f(-3) = -20 \)[/tex] (left neighbor)
- [tex]\( f(-1) = 0 \)[/tex] (right neighbor)
- [tex]\( -4 \)[/tex] is not greater than both [tex]\(-20\)[/tex] and [tex]\(0\)[/tex].
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 0 \)[/tex]:
- [tex]\( f(-2) = -4 \)[/tex] (left neighbor)
- [tex]\( f(0) = -2 \)[/tex] (right neighbor)
- [tex]\( 0 \)[/tex] is greater than both [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex], hence this is a local maximum.
- The interval around this maximum is [tex]\((-2, 0)\)[/tex].
- Checking other points, only [tex]\( x = -1 \)[/tex] satisfies the local maximum condition.
2. Local Minimum:
A local minimum occurs where the value of [tex]\( f(x) \)[/tex] at a point is less than the values at neighboring points.
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -20 \)[/tex]:
- [tex]\( f(-4) = -54 \)[/tex] (left neighbor)
- [tex]\( f(-2) = -4 \)[/tex] (right neighbor)
- [tex]\( -20 \)[/tex] is not less than both [tex]\(-54\)[/tex] and [tex]\(-4\)[/tex].
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -4 \)[/tex]:
- [tex]\( f(-3) = -20 \)[/tex] (left neighbor)
- [tex]\( f(-1) = 0 \)[/tex] (right neighbor)
- [tex]\( -4 \)[/tex] is not less than both [tex]\(-20\)[/tex] and [tex]\(0\)[/tex].
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -2 \)[/tex]:
- [tex]\( f(-1) = 0 \)[/tex] (left neighbor)
- [tex]\( f(1) = -4 \)[/tex] (right neighbor)
- [tex]\( -2 \)[/tex] is not less than both [tex]\(0\)[/tex] and [tex]\(-4\)[/tex].
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex]:
- [tex]\( f(0) = -2 \)[/tex] (left neighbor)
- [tex]\( f(2) = 0 \)[/tex] (right neighbor)
- [tex]\( -4 \)[/tex] is less than both [tex]\(-2\)[/tex] and [tex]\(0\)[/tex], hence this is a local minimum.
- The interval around this minimum is [tex]\((0, 2)\)[/tex].
3. Conclusion:
- Interval with local maximum: The interval [tex]\((-2, 0)\)[/tex] contains a local maximum.
- Interval with local minimum: The interval [tex]\((0, 2)\)[/tex] contains a local minimum.
These intervals indicate the ranges around the points where [tex]\( f(x) \)[/tex] attains local maxima and minima respectively.
Given the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & -54 & -20 & -4 & 0 & -2 & -4 & 0 & 16 & 50 \\ \hline \end{array} \][/tex]
We will look for the positions where [tex]\( f(x) \)[/tex] is greater than its neighbors (indicating a local maximum) or less than its neighbors (indicating a local minimum).
1. Local Maximum:
A local maximum occurs where the value of [tex]\( f(x) \)[/tex] at a point is greater than the values at neighboring points.
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -20 \)[/tex]:
- [tex]\( f(-4) = -54 \)[/tex] (left neighbor)
- [tex]\( f(-2) = -4 \)[/tex] (right neighbor)
- [tex]\( -20 \)[/tex] is not greater than both [tex]\(-54\)[/tex] and [tex]\(-4\)[/tex].
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -4 \)[/tex]:
- [tex]\( f(-3) = -20 \)[/tex] (left neighbor)
- [tex]\( f(-1) = 0 \)[/tex] (right neighbor)
- [tex]\( -4 \)[/tex] is not greater than both [tex]\(-20\)[/tex] and [tex]\(0\)[/tex].
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 0 \)[/tex]:
- [tex]\( f(-2) = -4 \)[/tex] (left neighbor)
- [tex]\( f(0) = -2 \)[/tex] (right neighbor)
- [tex]\( 0 \)[/tex] is greater than both [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex], hence this is a local maximum.
- The interval around this maximum is [tex]\((-2, 0)\)[/tex].
- Checking other points, only [tex]\( x = -1 \)[/tex] satisfies the local maximum condition.
2. Local Minimum:
A local minimum occurs where the value of [tex]\( f(x) \)[/tex] at a point is less than the values at neighboring points.
- At [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -20 \)[/tex]:
- [tex]\( f(-4) = -54 \)[/tex] (left neighbor)
- [tex]\( f(-2) = -4 \)[/tex] (right neighbor)
- [tex]\( -20 \)[/tex] is not less than both [tex]\(-54\)[/tex] and [tex]\(-4\)[/tex].
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -4 \)[/tex]:
- [tex]\( f(-3) = -20 \)[/tex] (left neighbor)
- [tex]\( f(-1) = 0 \)[/tex] (right neighbor)
- [tex]\( -4 \)[/tex] is not less than both [tex]\(-20\)[/tex] and [tex]\(0\)[/tex].
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -2 \)[/tex]:
- [tex]\( f(-1) = 0 \)[/tex] (left neighbor)
- [tex]\( f(1) = -4 \)[/tex] (right neighbor)
- [tex]\( -2 \)[/tex] is not less than both [tex]\(0\)[/tex] and [tex]\(-4\)[/tex].
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex]:
- [tex]\( f(0) = -2 \)[/tex] (left neighbor)
- [tex]\( f(2) = 0 \)[/tex] (right neighbor)
- [tex]\( -4 \)[/tex] is less than both [tex]\(-2\)[/tex] and [tex]\(0\)[/tex], hence this is a local minimum.
- The interval around this minimum is [tex]\((0, 2)\)[/tex].
3. Conclusion:
- Interval with local maximum: The interval [tex]\((-2, 0)\)[/tex] contains a local maximum.
- Interval with local minimum: The interval [tex]\((0, 2)\)[/tex] contains a local minimum.
These intervals indicate the ranges around the points where [tex]\( f(x) \)[/tex] attains local maxima and minima respectively.
