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. \(y=x^{4 / 3}\)

Short answer

The graph of the function \(y=x^{4 / 3}\) has an inflection point at \(x = 0\) but does not have any relative extrema.

Step-by-step solution

Graph the function

Input the function \(y=x^{4 / 3}\) into the graphing utility. You can use any utility that suits your needs but a popular option is desmos.com, a free online graphing calculator. With the function inputted, a graph will be generated

Identify inflection point

An inflection point occurs where the curve changes concavity. For the function \(y=x^{4 / 3}\), the graph changes from being concave down to being concave up at \(x=0\). This is the only inflection point present.

Confirm absence of relative extrema

An extrema occurs at the high or low points of the curve. For \(y=x^{4 / 3}\), there are no extrema because the function increases for both \(x > 0\) and \(x < 0\). Because there are no maximum or minimum points, there are no points of relative extrema.