Question

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.
the function has an absolute minimum at $x = c$ value but does not have an absolute maximum value on $(a,b)$
the function does not have any absolute extreme values on its domain
explain the results in terms of the extreme value theorem.
a. since the function $f$ is not continuous and the domain of $f$ is not a closed interval, $f$ may or may not attain any absolute extreme values on its domain
b. since the function $f$ is not continuous and the domain of $f$ is a closed interval, $f$ may or may not have any absolute extreme values on its domain
c. since the function $f$ is continuous on a closed interval, $f$ attains both an absolute maximum value and an absolute minimum value on its domain
d. since the function $f$ is continuous and the domain of $f$ is not a closed interval, $f$ may or may not have any absolute extreme values on its domain

Explanation:

The Extreme Value Theorem states that if a function \( f(x) \) is continuous on a closed interval \([m,n]\), then \( f(x) \) attains an absolute maximum and an absolute minimum on \([m,n]\). In this case, the domain \((a,b)\) is an open interval (not closed), and even though the function may be continuous on parts of the interval, the lack of a closed - interval domain means there are no guarantees about the existence of absolute extrema.

Answer:

D. Since the function \( f \) is continuous and the domain of \( f \) is not a closed interval, \( f \) may or may not have any absolute extreme values on its domain.