https://file.aiquickdraw.com/mathbot/file/1728967616025e7kdd46e.jpg

The problem asks you to determine whether the **Intermediate Value Theorem (IVT)** guarantees that the function \( k(x) = 2x^3 - 11x^2 + 6x + 6 \) 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 \( f(x) \) is continuous on a closed interval [a, b], and \( f(a) \) and \( f(b) \) have opposite signs (i.e., one is positive and the other is negative), then there must be at least one \( c \) in the interval such that \( f(c) = 0 \). In other words, the function must cross the x-axis at some point within the interval.

### Step-by-step explanation:

1. The problem specifically asks you to find values of \( k(1) \) and \( k(2) \) to see if they have opposite signs.
   
2. We are given \( k(x) = 2x^3 - 11x^2 + 6x + 6 \), so we can compute:
   \[
   k(1) = 2(1)^3 - 11(1)^2 + 6(1) + 6 = 2 - 11 + 6 + 6 = 3
   \]
   \[
   k(2) = 2(2)^3 - 11(2)^2 + 6(2) + 6 = 2(8) - 11(4) + 12 + 6 = 16 - 44 + 12 + 6 = -10
   \]
   
3. Since \( k(1) = 3 \) (positive) and \( k(2) = -10 \) (negative), they have opposite signs.

4. By the Intermediate Value Theorem, because the function \( k(x) \) is continuous on the interval [1, 2] and \( k(1) \) and \( k(2) \) 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:**
1. What are the exact conditions for the Intermediate Value Theorem to hold?
2. How can you check continuity for a given function?
3. What would happen if \( k(1) \) and \( k(2) \) had the same sign?
4. How would you approach other intervals like [2, 3] or [3, 4] in this problem?
5. Can the Intermediate Value Theorem tell you where exactly the zero is?

**Tip:** Always make sure the function is continuous before applying the IVT!

Determine whether the Intermediate Value Theorem guarantees that the function k(x) = 2x^3 - 11x^2 + 6x + 6 has a zero on the given interval.

Explore how to apply the Intermediate Value Theorem (IVT) to determine the existence of roots in the polynomial k(x) = 2x^3 - 11x^2 + 6x + 6. Learn to compute function values at specific points and confirm the existence of a zero on a given interval.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation