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7-53.

Use the first and second derivatives to determine the following locations for .

  1. Relative minima and maxima.

    Remember: Finding where or DNE will identify CANDIDATES for minima and maxima. You need to complete the 1st or 2nd Derivative Test to confirm which is which.

  2. Intervals at which is increasing and decreasing.

    When is positive, then has positive slopes, which means is increasing.

  3. Inflection points.

    Remember: Finding where or DNE will identify CANDIDATES for inflection points.
    You need to do further investigation to determine if it is (or is not) an inflection point.

    Either test for a sign change of before and after the candidate point. Or evaluate at the candidate.
    If (candidate) , then it is a point of inflection.

  4. Intervals at which is concave up and concave down.

    When is positive, then has positive slopes, which means is concave up.