Lesson 44.

NOTE: Be sure to read  Homework Format  and Homework Writing Policy BEFORE writing up your solutions to be turned in.
PROBLEMS WILL NOT BE GRADED AND NO CREDIT WILL BE GIVEN if these guidelines are not follows.

Due Friday, April 11

All functions considered below are assumed to have codomain equal to the set of real numbers, and for simplicity we will take the domain to be the same set, although much more generality is possible. It is assumed in this problem that you and your reader know what is meant by an “interval” and what intervals look like. For simplicity, we will consider only nonempty intervals.
Sect. 7:
  1. Let  J  be an interval and let  f  be a function. Give a precise algebraic definition of what it means to say that  f  is increasing on  J. 

  2. Let  J  be an interval, let  m  be a number, and let  f  be a function. Give a precise algebraic definition of what it means to say that  m  is a minimum value of  f  on  J.  (We are talking about “global” or “absolute” minimum here on the given interval, not local or relative minimum. We will also use the concept of maximum, but you don't have to define it explicitly.)

  3. Prove or disprove:
    If the function  f  is increasing on the interval  [0, 1],  then  f([0, 1])  = [ f(0), f(1)].
    Give a clear, complete proof (or disproof), using only simple algebra.

  4. Prove or disprove:
    If the function  f  has a minimum  m1  and a maximum  m2  on the interval  [0, 1],  then  f([0, 1])  = [m1, m2].
    Give a clear, complete proof (or disproof), using only simple algebra.

  5. Prove or disprove:
    If the function  f  has a negative minimum  m  on the interval  [0, 1],  and  f(0) = f(1) = 0,  then  [m, 0]  is a subset of  f([0, 1]).
    If you believe this is true, give a clear, complete proof. If you believe this is false, give a clear, complete counterexample, with explanation and graph. (You don't have to give the proof for the counterexample, but you should be able to prove it is a counterexample by using only simple algebra.)

NOTE: Be sure to read  Homework Format  and Homework Writing Policy BEFORE writing up your solutions to be turned in.
PROBLEMS WILL NOT BE GRADED AND NO CREDIT WILL BE GIVEN if these guidelines are not follows.


NOTE: As stated on the Course Home Page, all due dates are tentative. Assignments, or parts of assignments, may be postponed to a later date.


Last modified Apr 11, 2008 9:26 AM

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