To verify the Lagrange Mean Value Theorem (LMVT) for the function $f(x) = x^3 + x^2$ on the interval $[1, 2]$, let's proceed with the following steps:

1. Check if LMVT applies: LMVT requires that the function $f(x)$ is continuous on $[1, 2]$ and differentiable on $(1, 2)$. Since $f(x) = x^3 + x^2$ is a polynomial, it is both continuous and differentiable over all real numbers, so LMVT can be applied.

2. Compute $f(1)$ and $f(2)$: $f(1) = 1^3 + 1^2 = 1 + 1 = 2$ $f(2) = 2^3 + 2^2 = 8 + 4 = 12$

3. Calculate the average rate of change of $f(x)$ on $[1, 2]$: The average rate of change of $f(x)$ on $[1, 2]$ is given by $\frac{f(2) - f(1)}{2 - 1} = \frac{12 - 2}{1} = 10$

4. Find $f'(x)$: To apply LMVT, we need to find a point $c \in (1, 2)$ where $f'(c) = 10$. First, compute $f'(x)$: $f'(x) = \frac{d}{dx} (x^3 + x^2) = 3x^2 + 2x$

5. Set up the equation $f'(c) = 10$ and solve for $c$: Substitute $f'(c) = 10$ into the derivative: $3c^2 + 2c = 10$ Rearrange this equation to get: $3c^2 + 2c - 10 = 0$ Now, solve this quadratic equation using the quadratic formula: $c = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 3 \cdot (-10)}}{2 \cdot 3} = \frac{-2 \pm \sqrt{4 + 120}}{6} = \frac{-2 \pm \sqrt{124}}{6}$ Simplify $\sqrt{124}$ to $2\sqrt{31}$, so: $c = \frac{-2 \pm 2\sqrt{31}}{6} = \frac{-1 \pm \sqrt{31}}{3}$ We only take the positive root that lies within the interval $(1, 2)$: $c = \frac{-1 + \sqrt{31}}{3} \approx 1.62$

Thus, there exists a point $c \approx 1.62 \in (1, 2)$ where $f'(c) = 10$, satisfying the Lagrange Mean Value Theorem.

Would you like further details on any part of this verification?

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Related Questions

1. What are the main requirements for the Lagrange Mean Value Theorem to hold?
2. How does LMVT differ from the Rolle's Theorem?
3. Can the LMVT be applied to functions that are not differentiable?
4. How does the Mean Value Theorem relate to the concept of instantaneous rate of change?
5. In what other applications can we use the LMVT beyond polynomial functions?

Tip: For solving quadratic equations efficiently, always check if factoring is simpler than applying the quadratic formula!