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5-132.

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

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

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

2nd Derivative Test.

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

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

1st Derivative Test.

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

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

Two words.

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

Refer to hint in part (c).

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

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 2nd-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 3rd-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 is continuous at $x = 3$, but not differentiable there.

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

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

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 $x > 3$

Refer to hint in part (e).