If we are looking at the entire interval that we can see in the graph, the local maximum and local minimum are two points that are extrema within a specific interval. For this problem, it looks like we can assume that our interval is between -5 and 5 in both the x and y directions.
To find local extrema, we need to find the critical points of the graph. The critical points are the points where the derivative of the graph (rate of change) is 0. This is a point where the direction of the graph changes. It looks like there are two such points on the graph. The first is approximately (1, 3.66). At this point, the derivative becomes zero and the graph starts going down. Another such point occurs at approximately (4, -2.55). At this point, the derivative becomes zero again and then starts climbing. These are the only 2 DEFINITIVE critical points on the graph.
We have two critical points (1, 3.66) and (4, -2.55). Of these, the maximum is the one with the larger y-value. Therefore, Local maximum: approx. (1,3.66); local minimum: approx. (4,-2.55)