Question

the graph of a function f is given. use the graph to answer the question.

1. find the numbers, if any, at which f has a local minimum. what are the local minima?

Explanation:

Step1: Recall the definition of local minimum

A function \(y = f(x)\) has a local minimum at a point \(x = a\) if there exists an open interval \(I\) containing \(a\) such that \(f(a)\leq f(x)\) for all \(x\in I\).

Step2: Analyze the graph

Looking at the graph of the function \(y = f(x)\):

• At \(x=- 2\), if we consider a small open interval around \(x =-2\), the value of the function \(f(-2)=0\) is less than the values of the function for \(x\) in that open - interval.
• At \(x = 2\), if we consider a small open interval around \(x = 2\), the value of the function \(f(2)=0\) is less than the values of the function for \(x\) in that open - interval.

Answer:

The function \(f\) has local minima at \(x=-2\) and \(x = 2\). The local minima are \(f(-2)=0\) and \(f(2)=0\).