Clau
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PRÁCTICA DIRIGIDA Nº 1
2. x∈ [ -2,3 ]
m ≤ x+5/x+7 ≤M
-2
x≤3
5
x +7 ≤ 10
1/10
1/x+7
1/5
1/5 ≤ 2/x+7
2/5
-2/5 ≤ -2/x+7 3/5
→ m ≤ 1 - 2/x+7 ≤ M
-1/5
1- 2/x+7 ≤4/5
∴m = 3/5
3/5 ≤ x+5/x+7 ≤4/5
M = 4/5
4. X ∈ [ 1/2,3/2 ] 1/2
x
m
x+2/x-2 ≤M
3/2
-3/2 ≤ x-2 ≤ -1/2 -2 ≤ 1/x-2
-2/3
-8 ≤ 4/x-2
-8/3
-7 ≤ 1 + 4/x-2 -5/3 ∴ M - m = 16/3
15.
M = -5/3
m = -7
m ≤ 1 + 4/x-2 ≤ M
a)
1
1,4
∆ { 〈 2,6 〉 ∪ 〈 0,1 ] }
4
0
0 1
1
2 4
0
2
6
6
2
4
- 〈 2,4 〉
6 0,2
b)
0,6
→
∪ 〈 4,6 ]
[ 2,7 〉∆〈 0,1 〉} ∪ {{〈 -2,3 ]∩[ 2,6 ]©} ∪ { 〈 0,4 〉 ∩ {〈- ,2 〉∪ 〈 6,8 〉}}
0 1 2
-2 0
7
∪
1
-2
2
2 3 6
7
-2
0 2 4 6
∪
0
0 2 4
∪
1
2
〈 -2,7 〉
7
6
8
8
2. x∈ [ -2,3 ]
m ≤ x+5/x+7 ≤M
-2
x≤3
5
x +7 ≤ 10
1/10
1/x+7
1/5
1/5 ≤ 2/x+7
2/5
-2/5 ≤ -2/x+7 3/5
→ m ≤ 1 - 2/x+7 ≤ M
-1/5
1- 2/x+7 ≤4/5
∴m = 3/5
3/5 ≤ x+5/x+7 ≤4/5
M = 4/5
4. X ∈ [ 1/2,3/2 ] 1/2
x
m
x+2/x-2 ≤M
3/2
-3/2 ≤ x-2 ≤ -1/2 -2 ≤ 1/x-2
-2/3
-8 ≤ 4/x-2
-8/3
-7 ≤ 1 + 4/x-2 -5/3 ∴ M - m = 16/3
15.
M = -5/3
m = -7
m ≤ 1 + 4/x-2 ≤ M
a)
1
1,4
∆ { 〈 2,6 〉 ∪ 〈 0,1 ] }
4
0
0 1
1
2 4
0
2
6
6
2
4
- 〈 2,4 〉
6 0,2
b)
0,6
→
∪ 〈 4,6 ]
[ 2,7 〉∆〈 0,1 〉} ∪ {{〈 -2,3 ]∩[ 2,6 ]©} ∪ { 〈 0,4 〉 ∩ {〈- ,2 〉∪ 〈 6,8 〉}}
0 1 2
-2 0
7
∪
1
-2
2
2 3 6
7
-2
0 2 4 6
∪
0
0 2 4
∪
1
2
〈 -2,7 〉
7
6
8
8
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