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. \(y=x^{5 / 3}-5 x^{2 / 3}\)

Short answer

To graph and identify the key features of the function \(y=x^{5 / 3}-5 x^{2 / 3}\), input the function into a graphing utility and adjust the window as needed to clearly visualize the entire graph, paying special attention to its relative extrema and points of inflection.

Step-by-step solution

Understanding the function

The given function \(y=x^{5 / 3}-5 x^{2 / 3}\) is a rational exponent function. It's a combination of \(x^{5 / 3}\) and \(x^{2 / 3}\) terms. This function has relative extremum (maximum or minimum points) and points of inflection (where the curve changes concavity).

Plotting the function

To plot the function, first, key in the function equation into your graphing utility. This might differ based on the type of calculator or graphing utility software used. Most graphing utilities will start off with a standard window, often defined as −10 ≤ x ≤ 10 and −10 ≤ y ≤ 10.

Adjusting the window

If the default window option doesn't show the entire graph or the points of interest (relative extrema and inflection points), adjust the window settings. Visualize the graph properly to identify these points. It may take a bit of adjusting to get the window just right.

Identifying Relative Extrema and Points of Inflection

Once the function is properly graphed, identify the relative extrema (maximum and minimum points of the function) and points of inflection (where the function changes direction). You can use the graphing utility's built-in features to assist in this, refer the user manual or help guide if unsure how to use these features.