Final answer:
To approximate local maxima and minima for the function f(x) = -0.2x³ - 0.5x² + 4x - 6, one must graph the function, which typically involves graphing utilities. The function's curve will show peaks (maxima) and valleys (minima) at these points. The given coordinates require verification with an actual graph to confirm if they represent these local extrema.
Step-by-step explanation:
The student's question involves using a graphing utility to graph the function f(x) = -0.2x³ - 0.5x² + 4x - 6 and to approximate the local maxima and minima. To find these, one would typically use a graphing calculator or software to input the function and observe the graph. Local maxima and minima are the points on the graph where the curve changes direction from increasing to decreasing (maxima) or from decreasing to increasing (minima). However, without the graph, these points cannot be confirmed or denied. Therefore, it is critical to actually graph the function to identify these points accurately.
The given coordinates a) (2, -10), b) (1, -5), c) (-2, 10), and d) (-1, 5) may represent the local maxima and minima, but only a graphical representation will provide the information needed to determine if they do. Remember that local maxima will appear as peaks and local minima as valleys on the graph of the given cubic function.