Question

question
identify the local maximum and local minimum of the function shown in the graph below.

provide your answer below:

the local maximum is at ( , ) and the local minimum is at ( , )

Explanation:

Step1: Recall the definition of local maximum and minimum

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, we find the highest point in its local - neighborhood (local maximum) and the lowest point in its local - neighborhood (local minimum).
The local maximum occurs at the point \((3,1)\) as in the neighborhood of \(x = 3\), the function value \(y = 1\) is higher than the function values for \(x\) values close to \(3\).
The local minimum occurs at the point \((-1,-4)\) as in the neighborhood of \(x=-1\), the function value \(y = - 4\) is lower than the function values for \(x\) values close to \(-1\).

Answer:

The local maximum is at \((3,1)\) and the local minimum is at \((-1,-4)\)