Intermediate Value Theorem (IVT)
Why it matters
- The IVT tells us that continuous functions never skip values.
- It’s super useful for proving that equations have solutions (like showing a curve must cross the x-axis).
- It doesn’t give the exact answer but guarantees that the answer exists.
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The Idea (Super Simple)
Think of it like a journey:
- You start at f(a) = 500 miles.
- You end at f(b) = 1000 miles.
- If you travel smoothly (no teleporting), you must have been at 750 miles somewhere in between.
👉 That’s exactly what the IVT says: a continuous function has to pass through every number between its start and end values.
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The Formal Statement
If f(x) is continuous on a closed interval [a, b], and L is any value between f(a) and f(b), then there exists some number c in [a, b] such that:
f(c) = L
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Conditions
- Continuity: f must be continuous on [a, b].
- Closed interval: The interval includes both endpoints a and b.
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What the theorem guarantees
- Existence: It guarantees that at least one value c exists, but doesn’t tell you exactly where it is.
- Intermediate values: The function will take all values between f(a) and f(b).
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Example
- Suppose f(1) = -2 and f(3) = 4.
- Since the function is continuous, IVT says the curve must cross 0 somewhere between 1 and 3.
- That proves there’s a root of the equation f(x) = 0 in that interval.
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✅ In one line: A continuous function can’t jump — it must pass through every value between its start and end.