The first two problems deal with the following situation.
Suppose you throw a ball into the air, and at half-second intervals measure its height. Suppose this is the table of values you come up with.
| Time in Seconds | Height in Feet |
| 1/2 | 16 |
| 1 | 24 |
| 3/2 | 27 |
| 2 | 25 |
| 5/2 | 21 |
| 3 | 14 |
| 7/2 | 0 |
1. Did the ball ever reach a height of 12 feet? If so, when? How do you
know this?
2. Did the ball ever reach a height of 30 feet? If so, when? How do you
know this?
3. If f(1) = -2 and f(3) = 5, then what mst we know about the function f in
order to know that f(x) = 0 for some x between 1 and 3?
4. Let f(x) = x3 - 4x2 + 2x - 1. Here are some of its values already figured.
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| f(x) | -236 | -137 | -70 | -29 | -8 | -1 | -2 | -5 | -4 | 7 | 34 |
For how many values of x does f(x) = -3? Where are these x-values? How do
you know?
In each of the following problems, you are given a function and an interval. For each, determine whether or not the Extreme Value Theorem tells you that the function has a maximum and a minimum in the given interval. Explain why the Extreme Value Theorem does or does not apply.
| 5. f(x) = | 1/x | , [1,2] | | 7. | tan2(x) | , [0, /2) | | |
| 6. f(x) = | x3 + x2 - 5x + 3 | , [0,2] | | 8. | sqrt(x2 - 4) | , [2, 8] | | |
| x - 1 | 9. | ln(x2) | , [1, ) | |
10. (Hard Problem) Use the Extreme Value Theorem to help you prove the following:
x3 - 7x - 6 has a smallest value and a largest value on
the interval (-2,3).