Chapter 3: Worksheet 6 Jack K. Cohen Colorado School of Mines




Closed Interval Extremum Problem


Suggested Problems Section 3.5: 1, 3, 5, 9, 10, 12, 15, 19.




  1. The text states the important Maximum Value Property.
    1. State the corresponding Minimum Value Property and explain why it is true (assuming that the Maximum Value Property is true).
    2. State a combined theorem that covers both the Maximum and Minimum Value Properties.
    3. Give an example function and interval where the Maximum Value Property fails to hold because the function is discontinuous at a single point.
    4. Give an example function and interval where the Maximum Value Property fails to hold because the interval of continuity is not closed at one endpoint.
    5. Give an example function and interval where the Maximum Value Property does hold despite the fact that the function is discontinuous on the given interval.
    6. Give an example function and interval where the Maximum Value Property does hold despite the fact that the interval of continuity is not closed.
    7. Give an example where the hypotheses of the Maximum Value Property hold and the function attains its maximum at two points on the closed interval. Is there anything wrong with this?

  2. Give examples of the following situations or explain why the stated situation is impossible.
    1. f attains a minimum in [a, b], but does not attain a maximum there.
    2. f attains neither a maximum nor a minimum in [a, b].
    3. f is continuous on [a, b], but attains neither a maximum nor a minimum there.
    4. f attains a maximum in [a, b] at several points in this interval.

  3. (3.5.16)
    1. Find the maximum and minimum values attained by h(x) = x2 + 4x + 7 on the interval [- 3, 0].
    2. Check your answers intuitively using Plot.
    3. Explain why your answers are rigorously correct.

  4. (3.5.30) Find the maximum and minimum values attained by f (x) = | x + 1| + | x - 1| on the interval [- 2, 2]—also state the x values where these extrema are attained.

  5. Give examples of the following:
    1. f'(0) = 0, but f(0) is neither a local maximum nor a local minimum of f (x).
    2. f'(0)≠ 0, but f (0) is a local maximum of f (x).
    3. Suppose f'(0) = 0. State an intuitive condition on the values of f'(x) near x = 0 that guarantees that f (0) is a local maximum of f (x).
    4. Repeat last part for local minimum.