Question

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?
a. find the numbers, if any, at which f has a relative maximum, and find the relative maxima(maximum). select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the function f has (a) relative maxima(maximum) at 1 and the relative maxima(maximum) are(is) 4. (use a comma to separate answers as needed.)
b. the function f has no relative maxima.
b. find the numbers, if any, at which f has a relative minimum, and find the relative minima(minimum). select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the function f has (a) relative minima(minimum) at and the relative minima(minimum) are(is) . (use a comma to separate answers as needed.)
b. the function f has no relative minima.

Explanation:

Step1: Recall relative - maximum and minimum definitions

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 \(c\). A relative minimum occurs at a point \(x = c\) if \(f(c)\leq f(x)\) for all \(x\) in some open interval containing \(c\).

Step2: Analyze the graph for relative maxima

By observing the graph, we can see that the function reaches a peak at \(x = 1\) and the \(y\) - value at this peak is \(y = 4\). So the function has a relative maximum at \(x = 1\) and the relative - maximum value is \(4\).

Step3: Analyze the graph for relative minima

Upon observing the graph, we note that there are no points where the function has a local trough (a point lower than the surrounding points in an open interval). So the function has no relative minima.

Answer:

a. A. The function f has (a) relative maxima(maximum) at 1 and the relative maxima(maximum) are(is) 4.
b. B. The function f has no relative minima.