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. \(y=\frac{x^{2}}{x^{2}+3}\)

Short answer

The function \(y=\frac{x^{2}}{x^{2}+3}\) when graphed has no relative extrema and no points of inflection.

Step-by-step solution

Title: Input The Function

Firstly, open your graphing tool and input the function \(y=\frac{x^{2}}{x^{2}+3}\) into it. After you have entered the function, hit the enter key to graph the function.

Title: Choose a Suitable Window Size

Choose a window size that allows for all key features of the graph to be clearly visible. This includes the relative extrema and points of inflection. You can adjust your window size by zooming in or out as well as shifting the view up, down, left or right.

Title: Identify Relative Extrema

Look closely at the graph. Relative extrema are displayed as the peaks and valleys of the graph. The high point in a peak and the low point in a valley are relative extrema. For the function \(y=\frac{x^{2}}{x^{2}+3}\), there are no peaks nor valleys, implying no relative extrema.

Title: Identify Points of Inflection

Points of inflection are the points on the graph where the graph changes its curvature. For this function, there are no points of inflection as the graph doesn't change its curvature.