Mara Cook
2022-06-24
benedictazk
Beginner2022-06-25Added 22 answers
Your function has derivative which, as you say, is greater than 0. This means that is strictly increasing. This results, for instance, from the mean value theorem. Now suppose is 0 at two distinct points and , then . But we must have either or . We can assume and then since is increasing.
As you suggest, we use the intermediate value theorem to demonstrate that there is at least one solution to the problem. For any fixed we can choose large enough such that . If we choose large but negative we get . The intermediate value theorem now proves it.
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