The graph provided shows the function $f'$, which represents the derivative of some function $f$. We are asked to label the x-coordinates where the function $f$ has:

• Corners
• Cusps
• Horizontal tangents
• Vertical tangents

Here’s the analysis for each feature based on the graph of $f'$:

1. Corners: A corner occurs when the derivative $f'$ has a discontinuity but the function $f$ remains continuous. Corners typically show up where $f'$ changes abruptly but remains finite. From the graph, there's a clear corner at $x = -6$, as the slope of $f'$ changes abruptly.

2. Cusps: A cusp occurs when the derivative tends to infinity but in opposite directions as $x$ approaches a point. In this graph, there is a cusp at $x = 2$, where the graph of $f'$ shoots upwards, indicating a cusp.

3. Horizontal Tangents: A horizontal tangent occurs when the slope of $f'$ equals 0, which happens where $f' = 0$. This appears at $x = 0$, as the graph touches the x-axis, indicating a horizontal tangent.

4. Vertical Tangents: A vertical tangent happens when the derivative becomes infinite (or very steep). On the graph, this is seen at $x = 2$, where the slope of $f'$ becomes infinite.

Based on this analysis, here’s where the labels should go:

• Corner: At $x = -6$
• Cusp: At $x = 2$
• Horizontal Tangent: At $x = 0$
• Vertical Tangent: Also at $x = 2$ (because of the steep vertical rise in $f'$).

Would you like further clarification on any part of this, or the next steps for solving similar problems?

Here are 5 related questions to deepen your understanding:

1. How do cusps and corners differ in terms of the behavior of a function's derivative?
2. Why does a vertical tangent indicate that the derivative tends to infinity at a given point?
3. Can a function have both a vertical tangent and a cusp at the same point?
4. What does the presence of a horizontal tangent tell us about the original function $f$?
5. How would the graph of $f$ differ from the graph of $f'$ based on this information?

Tip: When analyzing the derivative graph $f'$, remember that horizontal tangents correspond to local maxima or minima on the original function $f$.