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.

To determine whether the Intermediate Value Theorem (IVT) applies to the function \( f(x) = x^2 - 4x \) over the interval \([6, 8]\), and whether it intersects the line \( y = 21 \) at least once within this interval, follow these steps:

### Step 1: Define the function that needs to be analyzed
Given:
\[
f(x) = x^2 - 4x
\]
We need to check if \( f(x) = 21 \) over the interval \([6, 8]\).

### Step 2: Define a new function to analyze the intersection
To use IVT, we consider the function:
\[
g(x) = f(x) - 21 = x^2 - 4x - 21
\]
We need to determine whether \( g(x) = 0 \) has a solution in the interval \([6, 8]\).

### Step 3: Evaluate the function at the endpoints of the interval
Calculate \( g(6) \) and \( g(8) \):
\[
g(6) = 6^2 - 4(6) - 21 = 36 - 24 - 21 = -9
\]
\[
g(8) = 8^2 - 4(8) - 21 = 64 - 32 - 21 = 11
\]

### Step 4: Check the conditions for the IVT
The Intermediate Value Theorem states that if \( g(x) \) is continuous on \([a, b]\) and \( g(a) \) and \( g(b) \) have opposite signs, then there exists at least one \( c \) in \((a, b)\) such that \( g(c) = 0 \).

Here:
- \( g(x) = x^2 - 4x - 21 \) is a polynomial function, which is continuous everywhere, including the interval \([6, 8]\).
- \( g(6) = -9 \) and \( g(8) = 11 \) have opposite signs.

Since \( g(x) \) is continuous on \([6, 8]\) and the values at the endpoints have opposite signs, by the IVT, there exists at least one \( c \in (6, 8) \) such that \( g(c) = 0 \).

### Step 5: Conclusion
The Intermediate Value Theorem applies, and it guarantees that there is at least one value \( c \) in the interval \((6, 8)\) such that \( f(c) = 21 \).

Would you like further details or have any questions? 

Here are some related questions to explore:
1. How do you solve \( x^2 - 4x - 21 = 0 \) using the quadratic formula?
2. How would you graph \( f(x) = x^2 - 4x \) and \( y = 21 \) on the same coordinate plane?
3. What is the geometric interpretation of the Intermediate Value Theorem?
4. How would you determine the exact value of \( c \) using numerical methods?
5. What happens if \( f(x) \) 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.

Sam is studying the function f(x) = x^2 − 4x. They want to prove that f(x) intersects the line y = 21 over the interval [6, 8] at least once. Determine whether IVT applies or not; if it applies, find the value of c that is guaranteed by the theorem.

Explore how to apply the Intermediate Value Theorem (IVT) to prove that the quadratic function f(x) = x^2 - 4x intersects the line y = 21 over the interval [6, 8]. This problem involves key algebraic concepts such as polynomial functions and continuity, suitable for students in Grades 10-12.

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