Question

(a) a function value f(a) is a local maximum value of f if f(a) is the value of f on some open interval containing a. from the graph of f we see that there are two local maximum the open interval (1, 7): one local maximum is, and it occurs when x = 2; the other loca largest smallest, and it occurs when x = (b) the function value f(a) is a local minimum value of f if f(a) is the value of f on some open interval containing a. from the graph of f we see that there is one local minimum value of f on the open interval (1, 7). the local minimum value is, and it occurs when x =

Explanation:

Step1: Recall local - maximum definition

A local maximum of a function \(y = f(x)\) is the largest value of \(f\) in some open - interval containing the point of interest.

Step2: Identify local maxima from the graph

Looking at the graph in the open interval \((1,7)\), when \(x = 2\), we can see that \(f(2)\) is a local maximum. Another local maximum occurs when \(x=5\). We estimate the \(y\) - values from the graph. Let's assume the \(y\) - value at \(x = 2\) is \(y_1\) and at \(x = 5\) is \(y_2\). By observing the graph, if the grid lines are of unit length, \(f(2)=5\) and \(f(5)=4\).

Step3: Recall local - minimum definition

A local minimum of a function \(y = f(x)\) is the smallest value of \(f\) in some open - interval containing the point of interest.

Step4: Identify local minimum from the graph

In the open interval \((1,7)\), the local minimum occurs at \(x = 3\), and by observing the graph, \(f(3)=2\).

Answer:

(a) largest; \(5\); \(2\); \(4\); \(5\)
(b) smallest; \(2\); \(3\)