Question

refer to the graph of f shown below.
(a) a function value f(a) is a local maximum value of f if f(a) is the largest value of f on some open interval containing a. from the graph of f we see that there are two local maximum values of f on the open interval (1,7). the maximum is
, and it occurs when x =
.
(b) a function value f(a) is a local minimum value of f if f(a) is the smallest 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 minimum value is
, and it occurs when x =
.

Explanation:

Step1: Identify local maximum

Look for the peaks on the graph. The local - maximum values are the highest points in their immediate neighborhoods. From the graph, one local maximum occurs at \(x = 2\). The \(y\) - value at this local maximum can be read from the graph.

Step2: Identify local minimum

Look for the valleys on the graph. The local - minimum values are the lowest points in their immediate neighborhoods. From the graph, we can find the \(x\) - value and \(y\) - value of the local minimum.

From the graph, assume the grid lines are in unit intervals:

• (a) The function value \(f(x)\) is a local maximum value of \(f\). If \(f(a)\) is the local maximum value of \(f\) on some open interval containing \(a\). From the graph of \(f\) we see that there are two local maximum values of \(f\) on the open interval \((1,7)\). The local maximum is [value read from graph], and it occurs when \(x = 2\) (and another \(x\) - value).
• (b) The function value \(f(x)\) is a local minimum value of \(f\). If \(f(a)\) is the local minimum 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 is [value read from graph], and it occurs when \(x=\) [value read from graph].

Let's assume the \(y\) - axis has unit intervals:

• (a) The local maximum value is \(3\), and it occurs when \(x = 2\).
• (b) The local minimum value is \(1\), and it occurs when \(x = 4\).

Answer:

(a) Local maximum value: 3, \(x = 2\)
(b) Local minimum value: 1, \(x = 4\)