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![Absolute Extrema on Closet Intervals [ a, b ]](https://image.slidesharecdn.com/5-131206202556-phpapp02-75d24f/85/5-4-absolute-maxima-and-minima-3-320.jpg)
![Find the absolute extrema of f (x) = 6x
4
3
-3x
1
3
on the interval [ -1, 1 ], and determine where
these values occur.
f '(x) = 8x
1
3
- x
-2
3
= x
-2
3
8x -1
( )
f '(x) =
8x -1
( )
x
2
3
Points to consider:
critical points
endpoints of the interval
1/8
0
-1
1
f (-1) = 9
f (0) = 0
f
1
8
脱
竪
巽
旦
淡
歎 = -
9
8
f (1) = 3
abs. max
abs. min](https://image.slidesharecdn.com/5-131206202556-phpapp02-75d24f/85/5-4-absolute-maxima-and-minima-4-320.jpg)
![How about some
practice?
1. f (x) =1+
1
x
Find the absolute extrema for the following:
0,+促
( )
2. f (x) = x3
e-2x 1,4
[ ]
3. f (x)=sinx-cosx 0,p
( ]](https://image.slidesharecdn.com/5-131206202556-phpapp02-75d24f/85/5-4-absolute-maxima-and-minima-9-320.jpg)
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- 1. Absolute Maxima and Minima
- 3. Absolute Extrema on Closet Intervals [ a, b ]
- 4. Find the absolute extrema of f (x) = 6x 4 3 -3x 1 3 on the interval [ -1, 1 ], and determine where these values occur. f '(x) = 8x 1 3 - x -2 3 = x -2 3 8x -1 ( ) f '(x) = 8x -1 ( ) x 2 3 Points to consider: critical points endpoints of the interval 1/8 0 -1 1 f (-1) = 9 f (0) = 0 f 1 8 脱 竪 巽 旦 淡 歎 = - 9 8 f (1) = 3 abs. max abs. min
- 5. Absolute Extrema on Infinite Intervals LIMITS CONCLUSION abs min no max abs max no min no max no min no max no min GRAPH lim x速-促 f (x) = +促 lim x速+促 f (x) = +促 lim x速-促 f (x) = -促 lim x速+促 f (x) = -促 lim x速-促 f (x) = -促 lim x速+促 f (x) = +促 lim x速-促 f (x) = +促 lim x速+促 f (x) = -促
- 6. Determine by inspection whether p(x) = 3x4 +4x3 has any absolute extrema. If so, find them and state where they occur. lim x速-促 3x4 +4x3 = +促 lim x速+促 3x4 +4x3 = +促 min exists p'(x)=12x3 +12x2 = 0 12x3 +12x2 = 0 12x2 x+1 ( )= 0 cp = -1,0 p(-1) = -1 p(0) = 0 abs min 揃
- 7. Absolute Extrema on Open Intervals LIMITS CONCLUSION abs min no max abs max no min no max no min no max no min GRAPH lim x速a+ f (x) = +促 lim x速b- f (x) = +促 lim x速a+ f (x) = -促 lim x速b- f (x) = -促 lim x速a+ f (x) = -促 lim x速b- f (x) = +促 lim x速a+ f (x) = +促 lim x速b- f (x) = -促
- 8. Determine by inspection whether p(x) = 1 x2 - x has any absolute extrema On the interval ( 0, 1 ). If so, find them and state where they occur. lim x速0+ 1 x2 - x = lim x速0+ 1 x x -1 ( ) = -促 max exists f '(x) = - 2x -1 x2 - x ( ) 2 = 0 cp = 0, 1 2 ,1 lim x速1- 1 x2 - x = lim x速1- 1 x x -1 ( ) = -促 f (0)= DNE f 1 2 脱 竪 巽 旦 淡 歎 = -4 f (1) = DNE 揃
- 9. How about some practice? 1. f (x) =1+ 1 x Find the absolute extrema for the following: 0,+促 ( ) 2. f (x) = x3 e-2x 1,4 [ ] 3. f (x)=sinx-cosx 0,p ( ]