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 ff that's continuous over the interval [a,b][a,b], the function will take any value between f(a)f(a) and f(b)f(b) over the interval.
More formally, it means that for any value LL between f(a)f(a) and f(b)f(b), there's a value cc in [a,b][a,b] for which f(c)=Lf(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))(a,f(a)) and (b,f(b))(b,f(b))...
... then it must pass through any yy-value between f(a)f(a) and f(b)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 ff with the following table of values. Let's find out where must there be a solution to the equation f(x)=2f(x)=2.
xx2-21-10011
f(x)f(x)44331-111
Note that f(1)=3f(-1)=3 and f(0)=1f(0)=-1. The function must take any value between 1-1 and 33 over the interval [1,0][-1,0].
22 is between 1-1 and 33, so there must be a value cc in [1,0][-1,0] for which f(c)=2f(c)=2.
Want to try more problems like this? Check out this exercise.
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