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.