To verify the Lagrange Mean Value Theorem (LMVT) for the function $f(x) = x^3 + x^2$ on the interval $[1, 2]$, we need to follow these steps:

1. Check if the function satisfies the conditions of LMVT:

   • The function $f(x) = x^3 + x^2$ is a polynomial, which means it is continuous on the closed interval $[1, 2]$ and differentiable on the open interval $(1, 2)$. So, the conditions are met.

2. Compute $f(a)$ and $f(b)$:

   • For $a = 1$ and $b = 2$: $f(1) = 1^3 + 1^2 = 1 + 1 = 2$ $f(2) = 2^3 + 2^2 = 8 + 4 = 12$

3. Apply the formula for LMVT: The theorem states that there exists a point $c \in (1, 2)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$ Substituting the values we found: $f'(c) = \frac{12 - 2}{2 - 1} = \frac{10}{1} = 10$

4. Calculate $f'(x)$ and find $c$:

   • Differentiate $f(x)$ to get $f'(x)$: $f'(x) = 3x^2 + 2x$
   • Set $f'(c) = 10$ and solve for $c$: $3c^2 + 2c = 10$ Rearranging, we get a quadratic equation: $3c^2 + 2c - 10 = 0$
   • Solving this quadratic equation using the quadratic formula: $c = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 3 \cdot (-10)}}{2 \cdot 3} = \frac{-2 \pm \sqrt{4 + 120}}{6} = \frac{-2 \pm \sqrt{124}}{6} = \frac{-2 \pm 2\sqrt{31}}{6}$ Simplifying further: $c = \frac{-1 \pm \sqrt{31}}{3}$
   • This yields two values for $c$, but we are interested in the one within the interval $(1, 2)$. Calculating: $c \approx \frac{-1 + \sqrt{31}}{3} \approx 1.62$

Thus, $c \approx 1.62$ is the point in the interval $(1, 2)$ where the derivative $f'(c) = 10$, confirming the Mean Value Theorem.

Would you like further details on any step, or do you have questions about the calculations?

Here are some additional questions related to this problem:

1. How does the Mean Value Theorem differ from Rolle's Theorem?
2. What would change if the interval were different, say $[0, 2]$?
3. Can we apply the Mean Value Theorem to functions that are not polynomials?
4. What is the geometric interpretation of the Mean Value Theorem?
5. How would the result change if the function were $f(x) = x^3 + 2x^2$?

Tip: Always ensure that the function is both continuous on $[a, b]$ and differentiable on $(a, b)$ before applying the Mean Value Theorem.