Question

for the graph of ( y = f(x) ) shown to the right, find the absolute minimum and the absolute maximum over the interval (1,11).

select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

a. the absolute minimum is at ( x = ) and ( x = )
(round to the nearest integer as needed. use ascending order.)

b. the absolute minimum is at ( x = )
(round to the nearest integer as needed.)

c. there is no absolute minimum.

Explanation:

Step1: Analyze the graph

We need to find the lowest \(y -\)value (absolute minimum) and the highest \(y -\)value (absolute maximum) on the interval \([1,11]\).
Looking at the graph, we observe the \(y -\)values of the function \(y = f(x)\) at different \(x -\)values in the interval \([1,11]\).

Step2: Determine the absolute minimum

The lowest \(y -\)value (absolute minimum) occurs at a single \(x -\)value in the interval \([1,11]\). By visual inspection of the graph, we can see that the function reaches its minimum value at \(x = 9\).

Step3: Determine the absolute maximum

The highest \(y -\)value (absolute maximum) occurs at \(x = 11\) (the right - hand endpoint of the interval \([1,11]\)) as the function is increasing towards \(x = 11\) within the given interval.

Answer:

The absolute minimum is \(7\) at \(x = 9\) and the absolute maximum is \(20\) at \(x = 11\) (assuming the \(y -\)values from the graph: if we count the grid - based on the general appearance of such problems, for the minimum at \(x = 9\) the \(y -\)value is around \(7\) and at \(x = 11\) the \(y -\)value is around \(20\)).