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. \(y=1-x^{2 / 3}\)

Short answer

The graph of the function \(y=1-x^{2 / 3}\) is U-shaped with an opening downwards. The global and local minimum is at the vertex of the curve. There are no points of inflection, as the curve does not change concavity.

Step-by-step solution

Understanding the Function

First, it's important to understand what kind of function you have here: \(y=1-x^{2 / 3}\). This is a power function with a negative sign in front of the variable term. Overall, the graph is expected to be U-shaped with the opening downwards because of the negative sign.

Using the Graphing Utility

Next, input the function \(y=1-x^{2 / 3}\) into the graphing utility. Depending on the particular utility, it's usually as simple as typing in the expression '1-x^(2/3)'.

Setting the Window

Now, adjust the window to view all the necessary points. Due to the fractional exponent, the graph is mostly defined for all real numbers, so set the window to include negative and positive values of 'x' as well as 'y'.

Identifying Relative Extrema and Inflection Points

Once the graph is displayed, look for any relative extrema – these are points on the graph where the function reaches a local minimum or maximum. In this case, the global minimum point also serves as a local minimum, which is the vertex of the u-shaped curve. For this function, there are no inflection points as the curve does not change concavity.