Suppose . In this problem, we will show that has exactly one root (or zero) in the interval .

(a) First, we show that has a root in the interval . Since is a function on the interval and and , the graph of must cross the -axis at some point in the interval by the . Thus, has at least one root in the interval .

(b) Second, we show that cannot have more than one root in the interval by a thought experiment. Suppose that there were two roots and in the interval with . Then . Since is on the interval and on the interval , by there would exist a point in interval so that . However, the only solution to is , which is not in the interval , since . Thus, cannot have more than one root in .

(Note: where the problem asks you to make a choice select the weakest choice that works in the given context. For example "continuous" is a weaker condition than "polynomial" because every polynomial is continuous but not vice-versa. Rolle's theorem is a weaker theorem than the mean value theorem because Rolle's theorem applies to fewer cases.)

You can earn partial credit on this problem.