Examine the mystery functions described below.

1. $f(x)$ has all of the following properties: $f^{\prime\prime}(2) < 0$, $f^\prime(2) = 0$ and $f(2) = 1$. Assuming the function is continuous, sketch a small portion of the mystery function near $x = 2$. Describe the function at this point.

   Hint (a):

   $f^\prime(2) = 0$ means that there COULD be a local maximum or local minimum at $x = 2$.
   $f^{\prime\prime}(2) < 0$ means that $f(x)$ is concave down at $x = 2$, confirming that $x = 2$ is a local ________________________.

2. A different mystery function has all of the following properties: $f^\prime(5) = 0$, $f^\prime(x) > 0$ for $4.98 < x < 5$, and $f^\prime(x) < 0$ for $5 < x < 5.02$. Draw a sketch of $f(x)$ near $x = 5$ and state a conclusion.

   Hint (b):

   If slope is positive on the left side of a point and negative on the right side, the that point is the location of a local ___________________.