Use the first and second derivatives to determine the following locations for $f(x) = xe^x$ .

1. Relative minima and maxima.

   Hint (a):

   Remember: Finding where $f^\prime(x) = 0$ or $f^\prime(x) =$ 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 $f(x)$ is increasing and decreasing.

   Hint (b):

   When $f^\prime(x)$ is positive, then $f(x)$ has positive slopes, which means $f(x)$ is increasing.

3. Inflection points.

   Hint (c):

   Remember: Finding where $f^{\prime\prime}(x) = 0$ or $f^{\prime\prime}(x) =$ DNE will identify CANDIDATES for inflection points.
   You need to do further investigation to determine if it is (or is not) an inflection point.

   Steps (c):

   Either test for a sign change of $f^{\prime\prime}(x)$ before and after the candidate point. Or evaluate $f^{\prime\prime}(x)$ at the candidate.
   If $f^{\prime\prime\prime}$(candidate) $≠ 0$, then it is a point of inflection.

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

   Hint (d):

   When $f^{\prime\prime}(x)$ is positive, then $f^\prime(x)$ has positive slopes, which means $f(x)$ is concave up.