Math Problem Statement
Sam is studying the function f(x)=x2−4xf(x)=x2−4x. They want to prove that f(x)f(x) intersects the line y=21y=21 over the interval [6,8][6,8] at least once. Determine whether IVT applies or not; if it applies, find the value of cc that is guaranteed by the theorem.
Solution
To determine whether the Intermediate Value Theorem (IVT) applies to the function over the interval , and whether it intersects the line at least once within this interval, follow these steps:
Step 1: Define the function that needs to be analyzed
Given: We need to check if over the interval .
Step 2: Define a new function to analyze the intersection
To use IVT, we consider the function: We need to determine whether has a solution in the interval .
Step 3: Evaluate the function at the endpoints of the interval
Calculate and :
Step 4: Check the conditions for the IVT
The Intermediate Value Theorem states that if is continuous on and and have opposite signs, then there exists at least one in such that .
Here:
- is a polynomial function, which is continuous everywhere, including the interval .
- and have opposite signs.
Since is continuous on and the values at the endpoints have opposite signs, by the IVT, there exists at least one such that .
Step 5: Conclusion
The Intermediate Value Theorem applies, and it guarantees that there is at least one value in the interval such that .
Would you like further details or have any questions?
Here are some related questions to explore:
- How do you solve using the quadratic formula?
- How would you graph and on the same coordinate plane?
- What is the geometric interpretation of the Intermediate Value Theorem?
- How would you determine the exact value of using numerical methods?
- What happens if were not continuous? Would the IVT still apply?
Tip: The Intermediate Value Theorem is a powerful tool for proving the existence of solutions within an interval, but it does not tell you exactly where the solution is—just that it exists.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Functions
Continuity
Formulas
f(x) = x^2 - 4x
g(x) = f(x) - 21
Theorems
Intermediate Value Theorem
Suitable Grade Level
Grades 10-12
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