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The image shows a math problem related to the Intermediate Value Theorem (IVT) and the polynomial function \( f(x) = 2x^3 + x - 2 \), over the interval \( (-1, 1) \). Let's go over the key parts of the solution:

1. **Continuity of the Function**:
   The function \( f(x) = 2x^3 + x - 2 \) is a polynomial. Polynomials are continuous for all real numbers, so the correct answer here is:
   - **B. It is continuous for all x.**

2. **Evaluating the Function at Endpoints**:
   - The value of the function at the left endpoint \( x = -1 \):
     \[
     f(-1) = 2(-1)^3 + (-1) - 2 = -2 - 1 - 2 = -5
     \]
     So, \( f(-1) = -5 \).
   
   - The value of the function at the right endpoint \( x = 1 \):
     \[
     f(1) = 2(1)^3 + 1 - 2 = 2 + 1 - 2 = 1
     \]
     So, \( f(1) = 1 \).

3. **Intermediate Value Theorem**:
   The IVT applies because:
   - The function \( f(x) = 2x^3 + x - 2 \) is continuous on \( [-1, 1] \).
   - The function takes values \( f(-1) = -5 \) and \( f(1) = 1 \), which means that since \( 0 \) is between \( -5 \) and \( 1 \), there must be some \( c \in (-1, 1) \) such that \( f(c) = 0 \).
   Thus, the correct answer is:
   - **C. It can be used because \( f(x) \) is continuous on \( [-1, 1] \) and \( 0 \) is between \( f(-1) \) and \( f(1) \).**

4. **Finding the Approximate Solution**:
   Using the IVT, the equation \( f(x) = 2x^3 + x - 2 = 0 \) has a solution in \( (-1, 1) \), which can be approximated through methods such as bisection or numerical solvers. To approximate \( x \), let me solve this for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can use numerical methods such as bisection or Newton's method to find an approximate solution to the equation \( 2x^3 + x - 2 = 0 \) within the interval \( (-1, 1) \).

Would you like to ask any questions or need further clarification? Here are some related questions that could expand your understanding:

1. What is the Intermediate Value Theorem, and why is continuity important for it?
2. How do you approximate the root of a function using the bisection method?
3. Can the Newton-Raphson method be applied here? If so, how would it work for this function?
4. What are the advantages and disadvantages of different numerical methods for root-finding?
5. How does polynomial continuity help in proving the existence of solutions?

**Tip**: When using numerical methods to find roots, always check the function's behavior at the interval's endpoints to ensure that the conditions of the theorem or method are satisfied.

2x^3 + x - 2 = 0 on the interval (-1,1). Evaluate the function at the endpoints and use the Intermediate Value Theorem to show the existence of a solution.

This problem involves applying the Intermediate Value Theorem to the polynomial function f(x) = 2x^3 + x - 2 on the interval (-1,1). The problem demonstrates the importance of continuity and shows how to find the solution using IVT.

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