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7-53.
  1. Use the first and second derivatives to determine the following locations for f(x) = xex . Homework Help ✎

    1. Relative minima and maxima.

    2. Intervals at which f(x) is increasing and decreasing.

    3. Inflection points.

    4. Intervals at which f(x) is concave up and concave down.

Remember: Finding where f '(x) = 0 or f '(x) = DNE will identify CANDIDATES for minima and maxima.
You need to complete the 1st or 2nd Derivative Test to confirm which is which.

When f '(x) is positive, then f(x) has positive slopes, which means f(x) is increasing.

Remember: Finding where f ''(x) = 0 or f ''(x) = 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 f ''(x) before and after the candidate point. Or evaluate f '''(x) at the candidate.
If f '''(candidate) ≠ 0, then it is a point of inflection.

When f ''(x) is positive, then f '(x) has positive slopes, which means f(x) is concave up.