Review the intermediate value theorem and use it to solve problems.
What is the intermediate value theorem?
The intermediate value theorem describes a key property of continuous functions: for any function f that's continuous over the interval [a,b], the function will take any value between f(a) and f(b) over the interval.
More formally, 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.
This theorem makes a lot of sense when considering the fact that the graphs of continuous functions are 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).
Want to learn more about the intermediate value theorem? Check out this video.
What problems can I solve with the intermediate value theorem?
Consider the continuous function f with the following table of values. Let's find out where must there be a solution to the equation f(x)=2.
Note that f(−1)=3 and f(0)=−1. The function must take any value between −1 and 3 over the interval [−1,0].
2 is between −1 and 3, so there must be a value c in [−1,0] for which f(c)=2.