Question

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 local - minimum definition

A local minimum of a function \(y = f(x)\) is a point where the function value is less than or equal to the values of the function in a small open - interval around that point. Visually, on a graph, it is a "valley" point.

Step2: Examine the graph

Looking at the given graph of the function \(y = f(x)\), we identify the points where the function has local minima.

Answer:

From the graph, the \(x\) - values at which \(f\) has local minima are \(x=- 4,x = 3\). The local minimum values of \(f\) are the \(y\) - values at these points. The local minimum value at \(x=-4\) is \(y = - 3\) and the local minimum value at \(x = 3\) is \(y=-1\). So the local minimum values of \(f\) are \(-3,-1\) and the values of \(x\) at which \(f\) has local minima are \(-4,3\).