4.6 Minimum and Maximum Values of Functions
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4.6Minimum and Maximum Values of Functions
Greg Kelly, Hanford High School, Richland, Washington
Borax Mine, Boron, CAPhoto by Vickie Kelly, 2004
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Extreme values can be in the interior or the end points of a function.
2y x
,D Absolute Minimum
No AbsoluteMaximum
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2y x
0,2D Absolute Minimum
AbsoluteMaximum
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2y x
0,2D No Minimum
AbsoluteMaximum
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2y x
0,2D No Minimum
NoMaximum
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Extreme Value Theorem:
If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.
Maximum & minimumat interior points
Maximum & minimumat endpoints
Maximum at interior point, minimum at endpoint
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Finding Maximums and Minimums Analytically:
1 Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points.
2 Find the value of the function at each critical point.
3 For closed intervals, check the end points as well.For open intervals, check the limits as x approaches the end points to determine if min or max exist within interval.
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EXAMPLE FINDING ABSOLUTE EXTREMA
Find the absolute maximum and minimum values of on the interval . 2/3f x x 2,3
2/3f x x
132
3f x x
3
23
f xx
There are no values of x that will makethe first derivative equal to zero.
The first derivative is undefined at x=0,so (0,0) is a critical point.
Because the function is defined over aclosed interval, we also must check theendpoints.
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2/ 3f x x 2,3D
At: 2x 232 2 1.5874f
At: 3x 233 3 2.08008f
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2/ 3f x x 2,3D
At: 2x 232 2 1.5874f
At: 3x 233 3 2.08008f
0 0f At: 0x
Absoluteminimum:
Absolutemaximum:
0,0
3,2.08
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EXAMPLE FINDING ABSOLUTE EXTREMA
Find the absolute maximum and minimum values of on interval . 2 3 1f x x x ,
2 3 1f x x x
2 3f x x Critical point at x = 1.5.
Because the function is defined over aopen interval, we also must check thelimits at the endpoints.
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21.5 1.5 3 1.5 1 3.25f
lim ( )x
f x
lim ( )x
f x
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Absoluteminimum:
Absolutemaximum:
1.5, 3.25
None
21.5 1.5 3 1.5 1 3.25f
lim ( )x
f x
lim ( )x
f x