determine from the given graph whether the function has any absolute extreme values on (a, b). then explain…

determine from the given graph whether the function has any absolute extreme values on (a, b). then explain how your answer is consistent with the extreme value theorem\ndetermine whether the function has any absolute extreme values on (a, b). choose the correct choice below\na. the function has an absolute maximum value at x = a but does not have an absolute minimum value on (a, b)\nb. the function has an absolute maximum value at x = a and an absolute minimum value at x = c on (a, b)\nc. the function has an absolute minimum at x = c value but does not have an absolute maximum value on (a, b)\nd. the function does not have any absolute extreme values on its domain.

determine from the given graph whether the function has any absolute extreme values on (a, b). then explain how your answer is consistent with the extreme value theorem\ndetermine whether the function has any absolute extreme values on (a, b). choose the correct choice below\na. the function has an absolute maximum value at x = a but does not have an absolute minimum value on (a, b)\nb. the function has an absolute maximum value at x = a and an absolute minimum value at x = c on (a, b)\nc. the function has an absolute minimum at x = c value but does not have an absolute maximum value on (a, b)\nd. the function does not have any absolute extreme values on its domain.

Answer

Explanation:

Step1: Analyze the graph

The Extreme Value Theorem states that if a function (y = f(x)) is continuous on a closed interval ([m,n]), then (f(x)) has both an absolute maximum and an absolute minimum on ([m,n]). But here the interval is ((a,b)) (open - interval). Looking at the graph of (y = f(x)):

  • There is no point in the open interval ((a,b)) where the function attains a minimum value. As we approach (x = a) from the right, the function values are higher than some values near (x=a) (but (x = a) is not in the open interval ((a,b))).
  • The function has a local maximum (and in this case, since there is no higher - valued point in the open interval ((a,b))) but no local minimum (and thus no absolute minimum) in the open interval ((a,b)).

Step2: Check each option

  • Option A: The function does not have an absolute maximum at (x = a) because (x=a) is not in the open interval ((a,b)).
  • Option B: (x = a) and (x = c) (if we assume (x = c) is a candidate). But (x=a) is not in the open interval ((a,b)). Also, there is no absolute minimum in the open interval ((a,b)) as we can get arbitrarily close to some lower - bound (but not reach it within the open interval).
  • Option C: There is no absolute minimum at (x = c) (since we can find values of the function less than (f(c)) as we approach (x=a) from the right within the open interval ((a,b))). And there is no absolute maximum (because the function is not defined on a closed interval where we could guarantee both extrema by the Extreme Value Theorem).
  • Option D: Since the interval ((a,b)) is open, and the function does not satisfy the conditions of the Extreme Value Theorem (which requires a closed and bounded interval for a continuous function to have both absolute extrema). A continuous function on an open interval may not have absolute extrema. In this case, there is no absolute minimum (we can approach some lower value but not attain it within ((a,b))) and no absolute maximum (if we consider the behavior near the endpoints of the open interval in the context of the graph shown, we can see that we can't pin - point an absolute maximum value within the open interval ((a,b)))

Answer:

D. The function does not have any absolute extreme values on its domain.