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. \(y=3 x^{2 / 3}-x^{2}\)

Short answer

Utilize a graphing tool to plot \(y=3 x^{2 / 3}-x^{2}\), identify the relative maxima and minima visually on the graph as your relative extrema, and find points of inflection where the curve changes from being concave up to down or vice versa.

Step-by-step solution

Graphing the Function

To start, we use a graphing utility to plot the function \(y=3 x^{2 / 3}-x^{2}\). Ensure to choose an appropriate scale that allows the entire function to be captured properly.

Identifying Relative Extrema

Following the plot, visually determine the highest and lowest points on the curve of the graph. These are the peaks and valleys referred as relative extrema. The highest points are known as relative maxima, while the lowest points represent relative minima.

Recognizing Points of Inflection

Next, we need to identify points of inflection which are points on the curve where it changes from being concave up to concave down or vice versa. The graph of our function will be concave up where the derivative is increasing and concave down where it is decreasing.