Question

refer to the graph of the function 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 shown above, we see that the leftmost local maximum value of f is 7 and that this value occurs when x is 5 (with checkmarks and xs). (b) the 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 shown above, we see that one local minimum value of f is _ and that this value occurs when x is _ (with xs).

Explanation:

Part (b)

Step1: Identify local minimum

A local minimum is a point where the function value is the smallest in its neighborhood. From the graph, we look for the lowest point (in its local interval). The graph has a local minimum, and we check the x - value where this occurs.

Step2: Determine x - value for local minimum

Looking at the graph, the local minimum (the smallest value in its open interval) occurs at \(x = 10\) (by observing the graph's curve, the lowest point in its local region is at \(x = 10\)). The value of \(f\) at this point can be seen from the graph's y - axis, but for the x - value of the local minimum, we identify the x - coordinate of the local minimum point.

Answer:

The x - value where the local minimum occurs is \(\boldsymbol{10}\) (assuming the graph's local minimum is at \(x = 10\), the value of \(f\) at this point can be determined from the graph's y - scale, but for the x - value, it is 10).