Let's solve the questions based on the graph of the derivative $f'$.

(a) Points of Inflection:

Points of inflection occur where the concavity of $f$ changes, which happens when $f''(x)$, the second derivative, changes sign. Since $f'(x)$ is the graph shown, the points of inflection of $f$ are where $f'(x)$ changes from increasing to decreasing or vice versa. This happens at local minima and maxima of the graph of $f'(x)$.

From the graph:

• Local maximum around $x = 1$
• Local minimum around $x = 4$

So, the points of inflection occur at $x = 1$ and $x = 4$.

Answer (a): 1, 4

(b) Concave Down Intervals:

To find where $f$ is concave down, we need to look for intervals where $f''(x)$ is negative, which corresponds to $f'(x)$ decreasing. This occurs where the slope of the graph of $f'(x)$ is negative.

From the graph, $f'(x)$ is decreasing between:

• $x \in (1, 4)$ (between the local maximum and minimum)

So, $f$ is concave down on the interval $(1, 4)$.

Answer (b): $(1, 4)$

Would you like further explanation or details about the solution? Here are some additional questions related to this problem:

1. What is the significance of a point of inflection in the context of real-world applications?
2. How can you determine if a function is concave up or down by just analyzing its second derivative?
3. What role do critical points play in determining local maxima or minima for a function?
4. Can a function have points of inflection without local maxima or minima in its derivative graph?
5. How do the first and second derivatives give us a complete picture of the behavior of a function?

Tip: Remember that concavity describes the "bend" of a graph. A concave-up graph looks like a "U", and a concave-down graph looks like an "n".