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.