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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?

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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.

Arrange the following numbers in increasing order: 0, g'(-2), g'(0), g'(2), g'(4).

This problem involves arranging the derivatives g'(-2), g'(0), g'(2), g'(4), and 0 based on the slope of the tangent lines from the graph of g(x). It covers key calculus concepts such as the derivative and its interpretation from a graph.

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