The problem asks to apply the Intermediate Value Theorem to the function $f(x) = x^3 + x$ over the interval $[1, 2]$, in order to find a value $c$ such that $f(c) = 9$.

Steps:

1. Check the function for continuity:

   • The function $f(x) = x^3 + x$ is a polynomial, and polynomials are continuous everywhere. Therefore, the function is continuous on the closed interval $[1, 2]$.

2. Evaluate $f(x)$ at the endpoints of the interval:

   • $f(1) = 1^3 + 1 = 2$
   • $f(2) = 2^3 + 2 = 8 + 2 = 10$

3. Apply the Intermediate Value Theorem (IVT):

   • The IVT states that if a function is continuous on a closed interval $[a, b]$ and $N$ is a number between $f(a)$ and $f(b)$, then there exists at least one $c \in [a, b]$ such that $f(c) = N$.
   • Here, we want to find $c$ such that $f(c) = 9$.
   • Since $f(1) = 2$ and $f(2) = 10$, and $9$ lies between $f(1)$ and $f(2)$, the IVT guarantees that there is a value $c \in [1, 2]$ such that $f(c) = 9$.

4. Solve for $c$ such that $f(c) = 9$:

   • Set up the equation: $c^3 + c = 9$
   • This is a cubic equation, and we can solve it numerically or graphically. The exact value of $c$ will lie between 1 and 2.

   To get an approximate value for $c$, we can estimate it numerically.

Would you like me to compute the value of $c$ more precisely? Let me know if you'd like further details on this step!

Here are some follow-up questions for a deeper understanding:

1. What is the precise solution of the equation $c^3 + c = 9$?
2. How can we solve cubic equations numerically?
3. Why is the Intermediate Value Theorem important in calculus?
4. What are other applications of the Intermediate Value Theorem?
5. Can we generalize this theorem to higher dimensions?

Tip: For polynomials, knowing the behavior of the function at critical points can help in understanding where specific values may lie.