Math Problem Statement
Solution
The problem asks you to determine whether the Intermediate Value Theorem (IVT) guarantees that the function has a zero on the given interval, using different intervals like [1, 2], [2, 3], etc.
The Intermediate Value Theorem (IVT)
The IVT states that if a function is continuous on a closed interval [a, b], and and have opposite signs (i.e., one is positive and the other is negative), then there must be at least one in the interval such that . In other words, the function must cross the x-axis at some point within the interval.
Step-by-step explanation:
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The problem specifically asks you to find values of and to see if they have opposite signs.
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We are given , so we can compute:
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Since (positive) and (negative), they have opposite signs.
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By the Intermediate Value Theorem, because the function is continuous on the interval [1, 2] and and have opposite signs, there is at least one zero of the function within this interval.
Let me know if you want to explore this further!
Questions to think about:
- What are the exact conditions for the Intermediate Value Theorem to hold?
- How can you check continuity for a given function?
- What would happen if and had the same sign?
- How would you approach other intervals like [2, 3] or [3, 4] in this problem?
- Can the Intermediate Value Theorem tell you where exactly the zero is?
Tip: Always make sure the function is continuous before applying the IVT!
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Math Problem Analysis
Mathematical Concepts
Intermediate Value Theorem
Polynomials
Continuity
Roots of Functions
Formulas
k(x) = 2x^3 - 11x^2 + 6x + 6
Theorems
Intermediate Value Theorem
Suitable Grade Level
Grades 10-12
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