To determine which intervals contain local maxima and minima for the given function based on the table, we will analyze the function values and their behavior around the points.
### Local Maximum
A local maximum is a point where the function value is greater than the values of its immediate neighbors. We can check the function values at each point and compare them with their adjacent points to find the local maximum.
We need to look at each [tex]\( f(x) \)[/tex] value and see if it is greater than the [tex]\( f(x) \)[/tex] values at positions immediately before and after it.
Let's check each interval step-by-step:
1. At [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -20, f(-4) = -54, f(-2) = -4 \][/tex]
Since [tex]\(-20\)[/tex] is not greater than both [tex]\(-54\)[/tex] and [tex]\(-4\)[/tex], it is not a maximum.
2. At [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -4, f(-3) = -20, f(-1) = 0 \][/tex]
Since [tex]\(-4\)[/tex] is not greater than both [tex]\(-20\)[/tex] and [tex]\(0\)[/tex], it is not a maximum.
3. At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 0, f(-2) = -4, f(0) = -2 \][/tex]
Since [tex]\(0\)[/tex] is greater than both [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex], this is a local maximum.
So, [tex]\( x = -1 \)[/tex] is a local maximum, and the interval surrounding it is [tex]\((-2, 0)\)[/tex].
### Local Minimum
A local minimum is a point where the function value is less than the values of its immediate neighbors. We can check the function values at each point and compare them with their adjacent points to find the local minimum.
We need to look at each [tex]\( f(x) \)[/tex] value and see if it is less than the [tex]\( f(x) \)[/tex] values at positions immediately before and after it.
Let's check each interval step-by-step:
1. At [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -20, f(-4) = -54, f(-2) = -4 \][/tex]
Since [tex]\(-20\)[/tex] is not less than both [tex]\(-54\)[/tex] and [tex]\(-4\)[/tex], it is not a minimum.
2. At [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -4, f(-3) = -20, f(-1) = 0 \][/tex]
Since [tex]\(-4\)[/tex] is not less than both [tex]\(-20\)[/tex] and [tex]\(0\)[/tex], it is not a minimum.
3. At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 0, f(-2) = -4, f(0) = -2 \][/tex]
Since [tex]\(0\)[/tex] is not less than both [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex], it is not a minimum.
4. At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2, f(-1) = 0, f(1) = -4 \][/tex]
Since [tex]\(-2\)[/tex] is not less than both [tex]\(0\)[/tex] and [tex]\(-4\)[/tex], it is not a minimum.
5. At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -4, f(0) = -2, f(2) = 0 \][/tex]
Since [tex]\(-4\)[/tex] is less than both [tex]\(-2\)[/tex] and [tex]\(0\)[/tex], this is a local minimum.
So, [tex]\( x = 1 \)[/tex] is a local minimum, and the interval surrounding it is [tex]\((0, 2)\)[/tex].
### Final Answer
- The interval containing a local maximum for this function is [tex]\((-2, 0)\)[/tex].
- The interval containing a local minimum for this function is [tex]\((0, 2)\)[/tex].
