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Questions and Answers
Which condition is necessary for the Intermediate Value Theorem to apply?
Which condition is necessary for the Intermediate Value Theorem to apply?
- The function must be increasing on [a, b]
- The function must be bounded on [a, b]
- The function must be continuous on [a, b] (correct)
- The function must be differentiable on [a, b]
What does it imply if $f(a) > 0$ and $f(b) < 0$ for a continuous function $f(x)$ on [a, b]?
What does it imply if $f(a) > 0$ and $f(b) < 0$ for a continuous function $f(x)$ on [a, b]?
- The function has a maximum at x = a
- There is at least one zero in the interval [a, b] (correct)
- The function is increasing on [a, b]
- The function has a minimum at x = 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?
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?
- The function must be constant
- The theorem cannot be applied (correct)
- There is no zero in the interval [a, b]
- The function must have a zero in the interval
If the function $f(x)$ is continuous at only some points in [a, b], what can be said about the Intermediate Value Theorem?
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?
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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