Let's analyze the graph and solve the questions based on the provided graph of the derivative $f'(x)$.

1. Where is $f(x)$ increasing?
   A function is increasing when its derivative is positive. From the graph, $f'(x) > 0$ between approximately $(3, 5)$. Thus, the answer for this part is: $(3, 5)$

2. Where is $f(x)$ concave down?
   A function is concave down where its second derivative is negative, which corresponds to intervals where the slope of $f'(x)$ is decreasing. In other words, $f(x)$ is concave down where $f'(x)$ has a negative slope. Based on the graph, the concave down interval seems to be approximately from $(1, 4)$. So, the answer is: $(1, 4)$

3. Local Minima:
   Local minima occur where $f'(x)$ crosses the x-axis from negative to positive (i.e., where $f'(x) = 0$ and changes from negative to positive). From the graph, this happens at approximately $x = 3$, so the local minimum is: $3$

4. Inflection Points:
   Inflection points occur where the concavity changes, which corresponds to where $f'(x)$ has local maxima or minima (i.e., where the slope of $f'(x)$ changes sign). From the graph, inflection points are at approximately $x = 1$, $x = 4$, $x = 5$, and $x = 6$. Thus, the inflection points are: $1, 4, 5, 6$

Would you like more details on any part of the solution or further explanation? Here are 5 related questions for deeper understanding:

1. What is the relationship between the derivative of a function and the concavity of the original function?
2. How can you determine if a critical point is a local minimum, maximum, or inflection point using the second derivative test?
3. What does it mean for a function to have a local maximum?
4. Why do we look at where the derivative changes sign to find inflection points?
5. How does the behavior of $f'(x)$ near an inflection point help us understand the graph of $f(x)$?

Tip: Inflection points are not just where the second derivative equals zero, but where the sign of the concavity changes!