At which point(s) on $f\left(x\right)$, graph below, do the following situations occur?

1. $f ^\prime\left(x\right)$ and $f ^{\prime\prime}\left(x\right)$ are both positive

   Hint (a):

   Translation: slope of $f\left(x\right)$ is positive AND $f\left(x\right)$ is concave up.

2. $f^\prime(x)$ and $f ^{\prime\prime}\left(x\right)$ are both negative

   Hint (b):

   Refer to the hint in part (a).

3. $f ^\prime\left(x\right)$ is positive and $f ^{\prime\prime}\left(x\right)$ is negative

   Hint (c):

   Refer to the hint in part (a).

4. $f ^\prime\left(x\right)$ is negative and $f ^{\prime\prime}\left(x\right)$ is positive

   Answer (d):

   Points $e$ and $a$.