To determine the intervals containing a local maximum and a local minimum, we need to inspect the given function values [tex]\( f(x) \)[/tex] at different [tex]\( x \)[/tex] points and compare them with their neighboring values.
Here is the step-by-step process for finding a local maximum:
1. Identify the points of interest: We need to compare each value [tex]\( f(x_i) \)[/tex] with its neighbors [tex]\( f(x_{i-1}) \)[/tex] and [tex]\( f(x_{i+1}) \)[/tex].
2. Compare values to find local maxima:
- [tex]\( f(-3) = -20 \)[/tex] is greater than [tex]\( f(-4) = -54 \)[/tex] but not greater than [tex]\( f(-2) = -4 \)[/tex].
- [tex]\( f(-2) = -4 \)[/tex] is greater than [tex]\( f(-3) = -20 \)[/tex] and greater than [tex]\( f(-1) = 0 \)[/tex].
- [tex]\( f(-1) = 0 \)[/tex] is equal to [tex]\( f(1) = 0 \)[/tex] and greater than [tex]\( f(0) = -2 \)[/tex].
- [tex]\( f(3) = 16 \)[/tex] is not greater than [tex]\( f(2) = 0 \)[/tex] but not greater than [tex]\( f(4) = 50 \)[/tex].
The point [tex]\( f(-2) = -4 \)[/tex] is surrounded by the values [tex]\( f(-3) = -20 \)[/tex] and [tex]\( f(-1) = 0 \)[/tex], making [tex]\([-2, 0]\)[/tex] the interval containing a local maximum for this function.
Now, to find a local minimum, we need to follow similar steps:
1. Identify the points of interest: Again, we compare each value [tex]\( f(x_i) \)[/tex] with its neighbors [tex]\( f(x_{i-1}) \)[/tex] and [tex]\( f(x_{i+1}) \)[/tex].
2. Compare values to find local minima:
- [tex]\( f(0) = -2 \)[/tex] is lesser than [tex]\( f(-1) = 0 \)[/tex] and lesser than [tex]\( f(1) = -4 \)[/tex].
- [tex]\( f(1) = -4 \)[/tex] is lesser than [tex]\( f(0) = -2 \)[/tex] but not greater than [tex]\( f(2) = 0 \)[/tex].
Thus, the point [tex]\( f(1) = -4 \)[/tex] is surrounded by the values [tex]\( f(0) = -2 \)[/tex] and [tex]\( f(2) = 0 \)[/tex], marking [tex]\([0, 2]\)[/tex] as the interval containing a local minimum for this function.
Therefore, the intervals containing a local maximum and a local minimum for this function are:
- Local maximum interval: [tex]\((-2, 0)\)[/tex]
- Local minimum interval: [tex]\((0, 2)\)[/tex]
