Closed Intervals

1. Block 1
Closed Intervals

2. What is to be learned?
• What a closed interval is
• How closed interval affects max or min
values
• Notation for closed intervals

3. Max
Min

4. Max
Min

5. Max
Min
Max/Min Values occur at
SPs OR end points

6. Closed intervals
• When there are limits on the domain
• Max or mins occur at SPs or End Points
Tactics → Find x values of SPS
Find corresponding y values
Repeat for end point x values
Find champ and chump
Notation x between -2 and 6 written as:
-2 ≤ x ≤ 6 or
[-2 , 6]

7. EX y = 1
/3 x3
– 9x for [-4 , 9]
SPs
for SPs dy
/dx = 0
dy
/dx = x2
– 9
x2
– 9 = 0
(x – 3)(x + 3) = 0
x = 3 or x = -3
No need to find nature!

8. y = 1
/3 x3
– 9x
contenders: SPs x = 3 and -3
End Points x = -4 and 9
x = 3 y = 1
/3 (3)3
– 9(3) =
x = -3 y = 1
/3 (-3)3
– 9(-3) =
x = -4 y = 1
/3 (-4)3
– 9(-4) =
x = 9 y = 1
/3 (9)3
– 9(9) =
Max Value = 216 at x = 9
Min Value = -18 at x = 3
- 18
18
14 2
/3
216

9. Find max and min values for
y = 2x3
– 6x2
for [-1 , 4]
SPs
for SPs dy
/dx = 0
dy
/dx = 6x2
– 12x
6x2
– 12x = 0
6x(x – 2) = 0
6x = 0 or x – 2 = 0
x = 0 or x = 2
x = 0, y = 0
x = 2, y = -8
x = -1, y = -8
x = 4, y = 34
max value = 34,
min = -8
Key
Question

10. Find max and min values for
y = 2x3
– 6x2
for [-1 , 4]
SPs
for SPs dy
/dx = 0
dy
/dx = 6x2
– 12x
6x2
– 12x = 0
6x(x – 2) = 0
6x = 0 or x – 2 = 0
x = 0 or x = 2
x = 0, y = 0
x = 2, y = -8
x = -1, y = -8
x = 4, y = 34
max value = 34,
min = -8
Key
Question