For each part below, what can you conclude (if anything) about $f(x)$ if you know the given information below? (Note: Each part is separate.)

1. $f^\prime(−2) = 0$ and $f^{\prime\prime}(−2) > 0$.

   Hint 1(a):

   This is a justification about whether $f(x)$ has a local max or a local min at $x = −2$. But which one?

   Hint 2(a):

   2nd Derivative Test.

2. $f^\prime(x) < 0$ when $x > 3$, $f^\prime(x) > 0$ when $x < 3$, and $f^\prime(3) = 0$.

   Hint 1(b):

   This is a justification about whether $f(x)$ has a local max or a local min at $x = 3$. But which one?

   Hint 2(b):

   $1$st Derivative Test.

3. $f^{\prime\prime}(3) > 0$

   Hint 1(c):

   Complete the sentence: At $x = 3$, the $2$nd derivative is positive so $f(3)$ is ____________________________

   Hint 2(c):

   Two words.

4. $f^{\prime\prime}(3) < 0$

   Hint (d):

   Refer to hint in part (c).

5. $f^{\prime\prime}(3) = 0$

   Answer (e):

   There might be a point of inflection on $f(x)$ at $x = 3$. But we do not know for sure. There are two ways to find out:
   1. You could check the $2$nd-derivative to the left of $x = 3$. If it changes signs then $f(3)$ is a POI, if not then it's not.
   2. You could evaluate the $3$rd-derivative at $x = 3$. If it is NOT zero, then $f(3)$ is a POI. If is is zero, then this method is inconclusive.
   Since we are not given any extra information, we can only say that $f(3)$ is a CANDIDATE for a point of inflection.

6. $f(x)$ is continuous at $x = 3$, but not differentiable there.

   Hint (f):

   Examples of points of NON-differentiability include: cusps, endpoints, jumps, holes and vertical tangents.
   Which of the above attributes still possess continuity?

7. $f(x)$ is defined and continuous everywhere, and has just one critical point at $x = 2$, which is a local maximum.

   Hint (g):

   Condition 1: Either $f^\prime(2) = 0$ or $f^\prime(2)$ does not exist.
   Condition 2: Refer to hints in part (a) or part (b).

8. $f^{\prime\prime}(3)$ does not exist and $f^{\prime\prime}(x) < 0$ for $x < 3$ and $f^{\prime\prime}(x) > 0$ for $x > 3$.

   Hint (h):

   Refer to hint in part (e).