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For each part below, what can you conclude (if anything) about f if you know the given information? (Note: Each part is different function.) Homework Help ✎

  1.  and

    This is a justification about whether f(x) has a local max or a local min at . But which one?

    2nd Derivative Test.

  2. This is a justification about whether has a local max or a local min at . But which one?

    1st Derivative Test.

  3. Complete the sentence: At , the 2nd derivative is positive so  is ____________________________

    Two words.

  4. Refer to hint in part (c).

  5. There might be a point of inflection on at . But we do not know for sure. There are two ways to find out:
    1. You could check the 2nd-derivative to the left of . If it changes signs then is a POI, if not then it's not.
    2. You could evaluate the 3rd-derivative at . If it is NOT zero, then is a POI. If is is zero, then this method is inconclusive.
    Since we are not given any extra information, we can only say that is a CANDIDATE for a point of inflection.

  6. f is continuous at , but not differentiable there.

    Examples of points of NON-differentiability include: cusps, endpoints, jumps, holes and vertical tangents.
    Which of the above attributes still possess continuity?

  7. f is defined and continuous everywhere, and has just one critical point at , which is a local maximum.

    Condition 1: Either or does not exist.
    Condition 2: Refer to hints in part (a) or part (b).

  8. does not exist and for and for

    Refer to hint in part (e).