To solve this problem, we need to analyze the graph and determine the relative values of the derivatives $f'(-4)$, $f'(-2)$, $f'(0)$, $f'(2)$, and $f'(4)$.

Approach:

1. Understanding the graph: The graph shows the function $f(x)$. The derivative values at specific points tell us the slope of the tangent line at those points.

2. Find slopes at specific points:

   • $f'(-4)$: At $x = -4$, the slope appears to be positive since the line is increasing.
   • $f'(-2)$: At $x = -2$, the graph is a peak, meaning the slope is zero here because it's a local maximum.
   • $f'(0)$: At $x = 0$, the graph is flat (horizontal), meaning the slope is zero again.
   • $f'(2)$: At $x = 2$, the graph is still increasing but flatter than at $x = -4$. The slope is slightly positive.
   • $f'(4)$: At $x = 4$, the graph is decreasing sharply, so the slope is negative.

Comparing the slopes:

• $f'(-4)$: Positive and relatively steep.
• $f'(-2)$: Zero (local maximum).
• $f'(0)$: Zero (flat section).
• $f'(2)$: Positive but not as steep as at $x = -4$.
• $f'(4)$: Negative and quite steep.

Ordering from least to greatest:

• The smallest slope is $f'(4)$ (negative value).
• Then we have $f'(-2)$ and $f'(0)$, both equal to zero.
• Next is $f'(2)$, a positive slope, but not as steep.
• Finally, the largest slope is $f'(-4)$, which is the steepest positive slope.

Thus, the order is: $f'(4) < f'(-2) = f'(0) < f'(2) < f'(-4)$

Let me know if you need any further clarification or details!

Here are five related questions you could explore:

1. What is the definition of the derivative and how does it relate to the slope of a graph?
2. What are critical points, and why do they occur at $x = -2$ and $x = 0$ on this graph?
3. How do you interpret increasing and decreasing intervals from a graph of a function?
4. How can you determine concavity from the graph of a function, and what does it say about the second derivative?
5. What is the significance of points where the derivative equals zero?

Tip: The derivative gives you the rate of change of a function. When analyzing graphs, local maximum and minimum points often correspond to where the derivative is zero, meaning the slope is flat.