We are asked to arrange the following numbers in increasing order: $0$, $g'(-2)$, $g'(0)$, $g'(2)$, and $g'(4)$. From the given graph of the function $g(x)$, let's analyze the slopes (derivatives) at the specified points, as the slope of the tangent line at any point $x$ is the derivative $g'(x)$.

Observing the graph:

• $g'(0)$: The slope of the tangent line at $x = 0$ appears to be zero, as the graph is flat at this point.
• $g'(2)$: The slope at $x = 2$ is positive, since the graph is increasing steeply around this point.
• $g'(4)$: The slope at $x = 4$ is also positive but less steep than at $x = 2$, so $g'(4) > g'(2)$, but both are positive.
• $g'(-2)$: The slope at $x = -2$ is negative, as the graph is decreasing steeply here.

Summary of slopes:

• $g'(0) = 0$ (flat slope)
• $g'(2) > 0$ (positive slope)
• $g'(4) > 0$ (positive slope, less than at $x = 2$)
• $g'(-2) < 0$ (negative slope)

Ordering:

The negative slope at $x = -2$ will be the smallest, followed by $g'(0) = 0$, then the positive slopes $g'(4)$ and $g'(2)$, with $g'(2)$ being steeper than $g'(4)$.

Thus, the correct increasing order is: $g'(-2), 0, g'(4), g'(2)$

Would you like more details, or have any questions?

Here are 5 follow-up questions related to this concept:

1. How can we determine the concavity of the graph based on the second derivative $g''(x)$?
2. What is the significance of a zero derivative at a point like $g'(0) = 0$?
3. How does the steepness of the graph relate to the magnitude of the derivative?
4. What does it mean if the derivative $g'(x)$ changes sign between points?
5. How can we estimate the exact value of the derivative from a graph?

Tip: The slope of the tangent line (derivative) at a point on a curve can give insight into whether the function is increasing, decreasing, or stationary at that point.