The intermediate value theorem focuses on a crucial part of continuity: for any function f(x) that's continuous over the interval [a, b], the function will take any y-value between f(a) and f(b) over the interval. 📈
Specifically, it means that for any value L between f(a) and f(b), there's a value c in [a,b] for which f(c) = L.
While this may sound confusing, it is actually very simple. As we have discussed before, continuous functions have to be drawn without lifting the pencil. If we know the graph passes through (a, f(a)) and (b, f(b)) then it must pass through any y-value between f(a) and f(b).
Image courtesy of Math is Fun
As you can see in the graph above, you can use the intermediate value theorem to prove that a function will cross a line at least once in an interval [a, b]. 👍
The IVT is a powerful tool, but it has its limitations. Take a look at this example from OSU about what the IVT can and cannot tell us.
In 2012, the Intermediate Value Theorem was the topic of an FRQ. Fin the full text of the prompt here courtesy of mathisfun.com. In summary, the Intermediate Value Theorem says that if a continuous function takes on two values y1 and y2 at points a and b, it also takes on every value between y1 and y2 at some point between a and b.
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