Question

here is a graph of the function f. use the graph to find the following. if there is more than one answer, separate them with commas. all values at which f has a local minimum: all local minimum values of f:

Explanation:

Step1: Recall the definition of local minimum

A local minimum of a function \(y = f(x)\) occurs at a point \(x = a\) if \(f(a)\) is less than or equal to the values of \(f(x)\) in some open interval containing \(a\). Visually, it is a "valley" on the graph.

Step2: Identify the \(x -\) values (local minimum points)

Looking at the graph, we observe two "valleys". The \(x -\) coordinates of these points are \(x=-3\) and \(x = 4\).

Step3: Identify the \(y -\) values (local minimum values)

The \(y -\) coordinate corresponding to \(x=-3\) is \(y = 1\), and the \(y -\) coordinate corresponding to \(x = 4\) is \(y=-2\).

Answer:

All values at which \(f\) has a local minimum: \(-3,4\)
All local minimum values of \(f\): \(1,-2\)