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Sagot :
To find the absolute extrema of the function [tex]\( f(x) = x^2 - 8x \)[/tex] on the given intervals, let's analyze each interval carefully.
### (a) Interval [tex]\([-1, 8]\)[/tex]
We need to find the minimum and maximum values of the function [tex]\( f(x) = x^2 - 8x \)[/tex] on the closed interval [tex]\([-1, 8]\)[/tex].
1. Evaluate the function at the endpoints:
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = (-1)^2 - 8(-1) = 1 + 8 = 9 \][/tex]
- At [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = 8^2 - 8(8) = 64 - 64 = 0 \][/tex]
2. Find the critical points by setting the derivative [tex]\( f'(x) = 0 \)[/tex]:
- The derivative [tex]\( f'(x) = 2x - 8 \)[/tex].
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 8 = 0 \implies x = 4 \][/tex]
- Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
- The point [tex]\( x=4 \)[/tex] lies within the interval [tex]\([-1, 8]\)[/tex].
3. Compare these values to determine the extrema:
- Minimum: [tex]\((4, -16)\)[/tex]
- Maximum: [tex]\((x,y)=(-1, 9)\)[/tex]
### (b) Interval [tex]\((4, 9]\)[/tex]
We need to find the extrema on the open interval [tex]\((4, 9]\)[/tex].
1. Evaluate the function at the endpoint [tex]\(x=9\)[/tex]:
[tex]\[ f(9) = 9^2 - 8(9) = 81 - 72 = 9 \][/tex]
2. Analyze the interval (4, 9]:
- The lower bound [tex]\(x = 4\)[/tex] is not included, so we cannot consider [tex]\(f(4)\)[/tex] for the minimum.
- The critical point [tex]\(x=4\)[/tex] lies within the interval but it is not included.
3. Determine the extrema:
- Minimum: Does Not Exist (DNE)
- Maximum: [tex]\( (9, 9) \)[/tex]
### (c) Interval [tex]\((0, 8)\)[/tex]
We need to find the extrema on the open interval [tex]\((0, 8)\)[/tex].
1. Evaluate the function at critical point [tex]\(x=4\)[/tex]:
[tex]\[ f(4) = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
2. Investigate boundary behavior (note that [tex]\(x=0\)[/tex] and [tex]\(x=8\)[/tex] are not included):
- [tex]\(f(x)\)[/tex] tends towards [tex]\(0\)[/tex] as [tex]\(x \)[/tex] approaches [tex]\(8\)[/tex] from within the interval.
3. Determine the extrema:
- Minimum: [tex]\( (4, -16) \)[/tex]
- Maximum: [tex]\( (0,0) \)[/tex]
### (d) Interval [tex]\([4, 10)\)[/tex]
We need to find the extrema on the interval [tex]\([4, 10)\)[/tex].
1. Evaluate the function at the endpoint [tex]\(x=4\)[/tex]:
[tex]\[ f(4) = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
2. Analyze the interval [4, 10):
- The upper bound [tex]\(x=10\)[/tex] is not included, so we can’t evaluate [tex]\(f(x)\)[/tex] at [tex]\( x=10 \)[/tex].
3. Determine the extrema:
- Minimum: [tex]\( (4, -16) \)[/tex]
- Maximum: Does Not Exist (DNE)
### (a) Interval [tex]\([-1, 8]\)[/tex]
We need to find the minimum and maximum values of the function [tex]\( f(x) = x^2 - 8x \)[/tex] on the closed interval [tex]\([-1, 8]\)[/tex].
1. Evaluate the function at the endpoints:
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = (-1)^2 - 8(-1) = 1 + 8 = 9 \][/tex]
- At [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = 8^2 - 8(8) = 64 - 64 = 0 \][/tex]
2. Find the critical points by setting the derivative [tex]\( f'(x) = 0 \)[/tex]:
- The derivative [tex]\( f'(x) = 2x - 8 \)[/tex].
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 8 = 0 \implies x = 4 \][/tex]
- Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
- The point [tex]\( x=4 \)[/tex] lies within the interval [tex]\([-1, 8]\)[/tex].
3. Compare these values to determine the extrema:
- Minimum: [tex]\((4, -16)\)[/tex]
- Maximum: [tex]\((x,y)=(-1, 9)\)[/tex]
### (b) Interval [tex]\((4, 9]\)[/tex]
We need to find the extrema on the open interval [tex]\((4, 9]\)[/tex].
1. Evaluate the function at the endpoint [tex]\(x=9\)[/tex]:
[tex]\[ f(9) = 9^2 - 8(9) = 81 - 72 = 9 \][/tex]
2. Analyze the interval (4, 9]:
- The lower bound [tex]\(x = 4\)[/tex] is not included, so we cannot consider [tex]\(f(4)\)[/tex] for the minimum.
- The critical point [tex]\(x=4\)[/tex] lies within the interval but it is not included.
3. Determine the extrema:
- Minimum: Does Not Exist (DNE)
- Maximum: [tex]\( (9, 9) \)[/tex]
### (c) Interval [tex]\((0, 8)\)[/tex]
We need to find the extrema on the open interval [tex]\((0, 8)\)[/tex].
1. Evaluate the function at critical point [tex]\(x=4\)[/tex]:
[tex]\[ f(4) = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
2. Investigate boundary behavior (note that [tex]\(x=0\)[/tex] and [tex]\(x=8\)[/tex] are not included):
- [tex]\(f(x)\)[/tex] tends towards [tex]\(0\)[/tex] as [tex]\(x \)[/tex] approaches [tex]\(8\)[/tex] from within the interval.
3. Determine the extrema:
- Minimum: [tex]\( (4, -16) \)[/tex]
- Maximum: [tex]\( (0,0) \)[/tex]
### (d) Interval [tex]\([4, 10)\)[/tex]
We need to find the extrema on the interval [tex]\([4, 10)\)[/tex].
1. Evaluate the function at the endpoint [tex]\(x=4\)[/tex]:
[tex]\[ f(4) = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
2. Analyze the interval [4, 10):
- The upper bound [tex]\(x=10\)[/tex] is not included, so we can’t evaluate [tex]\(f(x)\)[/tex] at [tex]\( x=10 \)[/tex].
3. Determine the extrema:
- Minimum: [tex]\( (4, -16) \)[/tex]
- Maximum: Does Not Exist (DNE)
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