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Sagot :
To determine the locations of the zeroes of the function [tex]\( I(x) \)[/tex], we need to look at the given data points and observe where the function value changes sign. The changes in sign indicate that the function crosses the x-axis between those points.
Let's analyze the data points step-by-step:
1. For [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex]:
- [tex]\( I(-4) = -221 \)[/tex] and [tex]\( I(-3) = 140 \)[/tex].
- The function changes from negative to positive, indicating a zero between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex].
2. For [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex]:
- [tex]\( I(-3) = 140 \)[/tex] and [tex]\( I(-2) = 231 \)[/tex].
- Both values are positive, no zero crossing here.
3. For [tex]\( x = -2 \)[/tex] and [tex]\( x = -1 \)[/tex]:
- [tex]\( I(-2) = 231 \)[/tex] and [tex]\( I(-1) = 160 \)[/tex].
- Both values are positive, no zero crossing here.
4. For [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex]:
- [tex]\( I(-1) = 160 \)[/tex] and [tex]\( I(0) = 35 \)[/tex].
- Both values are positive, no zero crossing here.
5. For [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
- [tex]\( I(0) = 35 \)[/tex] and [tex]\( I(1) = -36 \)[/tex].
- The function changes from positive to negative, indicating a zero between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex].
6. For [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
- [tex]\( I(1) = -36 \)[/tex] and [tex]\( I(2) = 55 \)[/tex].
- The function changes from negative to positive, indicating a zero between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].
7. For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
- [tex]\( I(2) = 55 \)[/tex] and [tex]\( I(3) = 416 \)[/tex].
- Both values are positive, no zero crossing here.
From this analysis, we find that there are zeroes:
- Between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex],
- Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex],
- Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].
Thus, the correct description of the zeroes is:
A. Between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex], between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex], and between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].
Let's analyze the data points step-by-step:
1. For [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex]:
- [tex]\( I(-4) = -221 \)[/tex] and [tex]\( I(-3) = 140 \)[/tex].
- The function changes from negative to positive, indicating a zero between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex].
2. For [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex]:
- [tex]\( I(-3) = 140 \)[/tex] and [tex]\( I(-2) = 231 \)[/tex].
- Both values are positive, no zero crossing here.
3. For [tex]\( x = -2 \)[/tex] and [tex]\( x = -1 \)[/tex]:
- [tex]\( I(-2) = 231 \)[/tex] and [tex]\( I(-1) = 160 \)[/tex].
- Both values are positive, no zero crossing here.
4. For [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex]:
- [tex]\( I(-1) = 160 \)[/tex] and [tex]\( I(0) = 35 \)[/tex].
- Both values are positive, no zero crossing here.
5. For [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
- [tex]\( I(0) = 35 \)[/tex] and [tex]\( I(1) = -36 \)[/tex].
- The function changes from positive to negative, indicating a zero between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex].
6. For [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
- [tex]\( I(1) = -36 \)[/tex] and [tex]\( I(2) = 55 \)[/tex].
- The function changes from negative to positive, indicating a zero between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].
7. For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
- [tex]\( I(2) = 55 \)[/tex] and [tex]\( I(3) = 416 \)[/tex].
- Both values are positive, no zero crossing here.
From this analysis, we find that there are zeroes:
- Between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex],
- Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex],
- Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].
Thus, the correct description of the zeroes is:
A. Between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex], between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex], and between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].
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