Download Math 131-Practice Quiz (4

“Big Theorems” Practice Quiz (4.8 &4.9)
Name:________________________________
Show your steps and justify your answers.
1. Without looking at your notes, state the following recent theorems:
a) The Intermediate Value Theorem:
b) The Mean Value Theorem:
2. Suppose f(x) = x3 – 5x +3
a) Must f reach a maximum value on the interval [-2, 1]? A minimum value on [-2, 1]?
Explain, using one of our named theorems.
b) Must f have a root in the interval [-2, 1]? Using one of our named theorems, prove
your answer is correct.
c) Does f satisfy (both) the hypotheses of the Mean Value Theorem on the interval
[-2,1]? If so, explain why and find all value(s) of the c mentioned in this theorem. If
not, explain why not.
3. Suppose f(x) = 1 – x2. Does f satisfy (both) the hypotheses of the MVT on [0, 3]? If so,
explain why and find all value(s) of the c mentioned in this theorem. If not, explain why not.
Draw a picture of f and the appropriate secant and tangent lines.
4. For each, tell whether such a function is possible or not. IF you say it is possible, give the
formula or graph of such a function. If you say it is not possible, carefully explain why not.
a) A function that is continuous on [1, 4] but which does not attain a minimum value on
[1, 4].
b) A function that attains a minimum value on [1,4] but that is not continuous on [1, 4].
c) A function f that is not continuous on [1, 4] but which takes on every y-value
between f(1) and f(4).