Question

the function f is given by the graph below.step 1 of 2: use the graph to determine the locations and type of the local extrema. write dne for all extrema that do not exist. separate multiple answers with a comma, if necessary.

Explanation:

Step1: Recall the definition of local maxima and minima

A local maximum is a point where the function changes from increasing to decreasing. A local minimum is a point where the function changes from decreasing to increasing.

Step2: Analyze the graph

Looking at the graph, at \(x = 0\), the function changes from increasing to decreasing. So, there is a local maximum at \(x = 0\). At some \(x\) - value (let's say \(x=a\)) where the function changes from decreasing to increasing. From the graph, if we assume the grid has unit - length squares, and by observing the turning points:
The function has a local maximum at \((0,0)\) (since at \(x = 0\), the \(y\) - value is \(0\) and the function changes its increasing - decreasing behavior) and a local minimum. If we assume the grid is such that the local minimum occurs at \(x = 2\) (by visual inspection of the graph's turning point). Let's assume the \(y\) - value at \(x = 2\) is \(- 1\) (based on the grid).

Answer:

Local maxima: \((0,0)\), Local minima: \((2, - 1)\)