To identify the local maxima and minima from the graph, we need to look for points where the function changes direction — specifically, where the slope of the graph changes from positive to negative (for a local maximum) or from negative to positive (for a local minimum).

Local Maxima:

A local maximum is a point where the function reaches a peak. The slope changes from positive (going up) to negative (going down) at this point.

• Around $x = -1$: The graph appears to have a peak around $x = -1$. This point is a local maximum because the graph rises before this point and falls after it. The approximate coordinates are $(-1, 2)$.

Local Minima:

A local minimum is a point where the function reaches a trough. The slope changes from negative (going down) to positive (going up) at this point.

• Around $x = -3$: The graph appears to have a trough around $x = -3$. This point is a local minimum because the graph falls before this point and rises after it. The approximate coordinates are $(-3, -2)$.

• Around $x = 1$: The graph also has a local minimum around $x = 1$. The graph falls before this point and rises afterward, making it a local minimum. The approximate coordinates are $(1, -3)$.

Summary:

• Local Maximum: $(-1, 2)$
• Local Minima: $(-3, -2)$ and $(1, -3)$

Would you like further details on how to interpret these points, or do you have additional questions about the graph?

Relative Questions:

1. How can you determine whether a critical point is a maximum or minimum just by looking at the graph?
2. What are the differences between absolute and local maxima/minima?
3. How would you find the x-intercepts on this graph?
4. Can a function have more than one local maximum or minimum?
5. What does the behavior of the function between the local maxima and minima tell you about the function's overall trend?

Tip:

When analyzing local maxima and minima, consider also looking at the first derivative (if available), as it gives a precise mathematical way to confirm these points.