Taylor uses Intermediate Value Theorem and concludes that the equation x^3−1x^2−7=K
 has a solution over the interval [1,7]
. Given their work is accurate, which of the following can be the value of K
?

The **Intermediate Value Theorem (IVT)** states that if a function \( f(x) \) is continuous on a closed interval \([a, b]\), and if \( f(a) \) and \( f(b) \) take opposite signs (or, more generally, if \( f(a) \) is less than some value \( K \) and \( f(b) \) is greater than \( K \), or vice versa), then there is at least one point \( c \in (a, b) \) such that \( f(c) = K \).

The given function is:

\[
f(x) = x^3 - x^2 - 7
\]

The task is to use the IVT to find a possible value of \( K \). 

### Step 1: Evaluate the function at the endpoints of the interval [1, 7].

- At \( x = 1 \):
  \[
  f(1) = 1^3 - 1^2 - 7 = 1 - 1 - 7 = -7
  \]

- At \( x = 7 \):
  \[
  f(7) = 7^3 - 7^2 - 7 = 343 - 49 - 7 = 287
  \]

### Step 2: Analyze the function's behavior.
Since \( f(1) = -7 \) and \( f(7) = 287 \), and the function is continuous (as it is a polynomial), by the Intermediate Value Theorem, any value of \( K \) between -7 and 287 can be a possible solution for the equation \( f(x) = K \) over the interval \([1, 7]\).

### Conclusion:
Given that \( K \) must lie within the range of the function over the interval \([1, 7]\), the value of \( K \) can be **any number between -7 and 287**. Therefore, any value within this range (e.g., \( K = 0 \), \( K = 50 \), etc.) is possible.

### Would you like more details or have any questions?

Here are 5 related questions for further understanding:
1. What are the conditions required for the Intermediate Value Theorem to apply?
2. Can the IVT guarantee more than one solution in this interval for \( f(x) = K \)?
3. What happens if the function was not continuous on the interval [1, 7]?
4. How can we verify the IVT for other types of functions?
5. How would you find the exact solution for \( f(x) = K \)?

**Tip:** Always check if the function is continuous before applying the IVT.

Taylor uses Intermediate Value Theorem and concludes that the equation x^3−1x^2−7=K has a solution over the interval [1,7]. Given their work is accurate, which of the following can be the value of K?

Explore how the Intermediate Value Theorem applies to find possible values of K for the polynomial function x^3 - x^2 - 7 over the interval [1,7]. Understand how the theorem ensures a solution within a given range. Suitable for students in Grades 10-12.

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