Question

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{-1 / 3}\)

Short answer

After graphing this function, you'll find out that it doesn't have any local minima or maxima. It has an inflection point at x = 0. For x>0, it curves upwards and for x<0, it curves downwards.

Step-by-step solution

Understanding the Function

Begin by understanding the function \(y=x^{-1/3}\), it is a root cubic function. This means that the function will have a graph that curves upwards for positive values of x and curves downwards for negative values of x.

Identify Extrema and Inflection Points

The function \(y=x^{-1/3}\) does not have any relative extrema because it curves upward and downward universally and does not attain a highest value in some interval (maximum) or lowest value in some interval (minimum). Therefore, it has no local maxima or minima. Also, function \(y=x^{-1/3}\) have a point of inflection at x = 0. As for x>0, the function curves upwards and for x<0, it curves downwards.

Choose Window for Graph

Choose a window which allows you to clearly see the properties of this function. A window from -10 to 10 on both axes would be suitable.

Graphing

Using a graphing utility, graph the function within the window specified. This will give a complete picture of how function behaves for both positive and negative values of x.