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Questions and Answers
Which condition is necessary for the Intermediate Value Theorem to apply?
What does it imply if $f(a) > 0$ and $f(b) < 0$ for a continuous function $f(x)$ on [a, b]?
In the context of the Intermediate Value Theorem, if $f(x)$ is continuous on [a, b] and $f(a)$ and $f(b)$ have the same sign, what can be concluded?
If the function $f(x)$ is continuous at only some points in [a, b], what can be said about the Intermediate Value Theorem?
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Which of the following is a correct application of the Intermediate Value Theorem?
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Study Notes
Intermediate Value Theorem
- The intermediate value theorem is applied to a function f(x)f(x)f(x) defined on the interval [a,b][a,b][a,b].
- The theorem requires the function to be continuous on the interval.
- If f(a)f(a)f(a) and f(b)f(b)f(b) have different signs, then the function has at least one zero in the interval [a,b][a,b][a,b].
- Geometrically, the theorem implies that the graph of the function crosses the xxx-axis at least once in the interval [a,b][a,b][a,b] when f(a)f(a)f(a) and f(b)f(b)f(b) have different signs.
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Description
The intermediate value theorem states that a continuous function with different signs at two endpoints has at least one zero in the interval.
