Question

in exercises 13 - 16, the graph of a function f is given. use the graph to find each of the following: a. the numbers, if any, at which f has a relative maximum. what are these relative maxima? b. the numbers, if any, at which f has a relative minimum. what are these relative minima?

Explanation:

Step1: Recall relative - maximum definition

A relative maximum of a function \(y = f(x)\) occurs at a point \(x = c\) if \(f(c)\geq f(x)\) for all \(x\) in some open interval containing \(c\). Visually, it is a peak on the graph.

Step2: Analyze the graph for relative maxima

For the graph of the function in question 13:

• We observe that the function has a relative maximum at \(x = 0\). The value of the function at \(x = 0\) is \(y = 4\). So the number at which \(f\) has a relative maximum is \(x = 0\) and the relative - maximum value is \(f(0)=4\).

Step3: Recall relative - minimum definition

A relative minimum of a function \(y = f(x)\) occurs at a point \(x = c\) if \(f(c)\leq f(x)\) for all \(x\) in some open interval containing \(c\). Visually, it is a valley on the graph.

Step4: Analyze the graph for relative minima

For the graph of the function in question 13:

• The function has relative minima at \(x=- 3\) and \(x = 3\). The value of the function at \(x=-3\) and \(x = 3\) is \(y = 0\). So the numbers at which \(f\) has relative minima are \(x=-3\) and \(x = 3\), and the relative - minimum values are \(f(-3)=0\) and \(f(3)=0\).

Answer:

a. The number at which \(f\) has a relative maximum is \(x = 0\), and the relative - maximum is \(4\).
b. The numbers at which \(f\) has relative minima are \(x=-3\) and \(x = 3\), and the relative - minima are \(0\).