Question

use the graph to determine the following. (a) find the numbers at which f has a relative maximum. what are these relative maxima? (b) find the numbers at which f has a relative minimum. what are these relative minima? (a) the number(s) at which f has a relative maximum is/are 3. (type an integer or a decimal. use a comma to separate answers as needed.) the relative maximum/maxima is/are 4. (type an integer or a decimal. use a comma to separate answers as needed.) (b) the number(s) at which f has a relative minimum is/are 2.4. (type an integer or a decimal. use a comma to separate answers as needed.) the relative minimum/minima is/are empty box. (type an integer or a decimal. use a comma to separate answers as needed.)

Explanation:

Step1: Recall relative - maximum concept

A relative maximum of a function occurs at a point where the function changes from increasing to decreasing. Looking at the graph visually, we identify the x - values where this happens.

Step2: Identify x - value for relative maximum

From the graph, we can see that the function changes from increasing to decreasing at \(x = 3\). So the number at which \(f\) has a relative maximum is \(x = 3\).

Step3: Find the relative - maximum value

We then find the corresponding y - value at \(x = 3\). By looking at the graph, when \(x = 3\), \(y=4\). So the relative maximum is \(4\).

Step4: Recall relative - minimum concept

A relative minimum of a function occurs at a point where the function changes from decreasing to increasing.

Step5: Identify x - values for relative minimum

From the graph, we can see that the function changes from decreasing to increasing at \(x = 2\) and \(x = 4\).

Step6: Find the relative - minimum values

By looking at the graph, when \(x = 2\) and \(x = 4\), the \(y\) - value is \(2\). So the relative minima are \(2\).

Answer:

(a) The number(s) at which \(f\) has a relative maximum is/are \(3\). The relative maximum/maxima is/are \(4\).
(b) The number(s) at which \(f\) has a relative minimum is/are \(2,4\). The relative minimum/minima is/are \(2\).