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# Intermediate value theorem review

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 open bracket, a, comma, b, close bracket, the function will take any value between f, left parenthesis, a, right parenthesis and f, left parenthesis, b, right parenthesis over the interval.
More formally, it means that for any value L between f, left parenthesis, a, right parenthesis and f, left parenthesis, b, right parenthesis, there's a value c in open bracket, a, comma, b, close bracket for which f, left parenthesis, c, right parenthesis, equals, 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 left parenthesis, a, comma, f, left parenthesis, a, right parenthesis, right parenthesis and left parenthesis, b, comma, f, left parenthesis, b, right parenthesis, right parenthesis...
... then it must pass through any y-value between f, left parenthesis, a, right parenthesis and f, left parenthesis, b, right parenthesis.

## 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, left parenthesis, x, right parenthesis, equals, 2.
xminus, 2minus, 101
f, left parenthesis, x, right parenthesis43minus, 11
Note that f, left parenthesis, minus, 1, right parenthesis, equals, 3 and f, left parenthesis, 0, right parenthesis, equals, minus, 1. The function must take any value between minus, 1 and 3 over the interval open bracket, minus, 1, comma, 0, close bracket.
2 is between minus, 1 and 3, so there must be a value c in open bracket, minus, 1, comma, 0, close bracket for which f, left parenthesis, c, right parenthesis, equals, 2.
Problem 1
f is a continuous function.
f, left parenthesis, minus, 2, right parenthesis, equals, 3 and f, left parenthesis, 1, right parenthesis, equals, 6.
Which of the following is guaranteed by the Intermediate Value Theorem?