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

Short answer

The function \(y=x^{1/3}+1\) has no extrema or inflection points. The graph is a smooth, rising curve that passes through the point (0,1) and extends from negative to positive infinity in both the x and y directions.

Step-by-step solution

Input Function

To graph the function, first input the function \(y=x^{1/3}+1\) into a graphing utility.

Choose Appropriate Window

Next, choose a window that allows all relative extrema and inflection points to be shown. Since the function given is a cube root function, the graph of the function will extend from negative to positive infinity in both the x and y directions. So, a wide window is recommended, such as x: [-10, 10] and y: [-10, 10].

Identify Extrema and Inflection Points

The function \(y=x^{1/3}+1\) has neither relative extrema nor inflection points. The graph has a smooth curve but it does not turn, meaning no extrema (minimum or maximum points) or points of inflection (points where the curve changes concavity) are present. Hence, the final graph should resemble a smoothly increasing curve that emerges from the negative x-axis, passes through the points (0,1) and continues to rise along the positive x-axis. Note that the '1' in the given function shifts the graph upward by one unit.