Key Concepts and Formulas

• Lagrange's Mean Value Theorem (LMVT): If a function $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.
• Derivative of a polynomial: $\frac{d}{dx}(x^n) = nx^{n-1}$
• Quadratic Formula: For a quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Step-by-Step Solution

Step 1: Verify the conditions of LMVT

• What: Check if the given function $f(x) = x^3 - 4x^2 + 8x + 11$ satisfies the conditions of LMVT on the interval $[0, 1]$.
• Why: LMVT can only be applied if the function is continuous on the closed interval and differentiable on the open interval.
• How: Polynomial functions are continuous and differentiable everywhere. Thus, $f(x)$ is continuous on $[0, 1]$ and differentiable on $(0, 1)$.

Step 2: Calculate $f(a)$ and $f(b)$

• What: Evaluate the function at the endpoints of the interval, $a = 0$ and $b = 1$.
• Why: These values are needed to compute the average rate of change $\frac{f(b) - f(a)}{b - a}$.
• How: $f(0) = (0)^3 - 4(0)^2 + 8(0) + 11 = 11$ $f(1) = (1)^3 - 4(1)^2 + 8(1) + 11 = 1 - 4 + 8 + 11 = 16$

Step 3: Calculate $f'(x)$

• What: Find the derivative of the function $f(x)$.
• Why: The derivative is needed to find $f'(c)$, which is used in the LMVT formula.
• How: $f'(x) = \frac{d}{dx}(x^3 - 4x^2 + 8x + 11) = 3x^2 - 8x + 8$

Step 4: Apply LMVT

• What: Substitute the calculated values into the LMVT formula: $f'(c) = \frac{f(b) - f(a)}{b - a}$.
• Why: This will give us an equation to solve for $c$.
• How: $f'(c) = 3c^2 - 8c + 8$ $\frac{f(1) - f(0)}{1 - 0} = \frac{16 - 11}{1} = 5$ $3c^2 - 8c + 8 = 5$

Step 5: Solve for $c$

• What: Solve the quadratic equation $3c^2 - 8c + 3 = 0$ for $c$.
• Why: The solutions to this equation are the possible values of $c$ that satisfy the LMVT.
• How: Use the quadratic formula: $c = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(3)}}{2(3)} = \frac{8 \pm \sqrt{64 - 36}}{6} = \frac{8 \pm \sqrt{28}}{6} = \frac{8 \pm 2\sqrt{7}}{6} = \frac{4 \pm \sqrt{7}}{3}$ So, $c = \frac{4 + \sqrt{7}}{3}$ or $c = \frac{4 - \sqrt{7}}{3}$.

Step 6: Check if $c$ lies in the interval $(0, 1)$

• What: Determine which of the two values of $c$ lies within the open interval $(0, 1)$.
• Why: LMVT requires $c$ to be strictly within the interval $(a, b)$.
• How: Since $\sqrt{7} \approx 2.646$, $c_1 = \frac{4 + \sqrt{7}}{3} \approx \frac{4 + 2.646}{3} \approx \frac{6.646}{3} \approx 2.215 > 1$ $c_2 = \frac{4 - \sqrt{7}}{3} \approx \frac{4 - 2.646}{3} \approx \frac{1.354}{3} \approx 0.451$ Since $0 < 0.451 < 1$, $c_2$ lies in the interval $(0, 1)$.

However, the provided answer is $2/3$. Let's re-examine the quadratic equation solution.

$3c^2 - 8c + 3 = 0$ $c = \frac{8 \pm \sqrt{64 - 36}}{6} = \frac{8 \pm \sqrt{28}}{6} = \frac{8 \pm 2\sqrt{7}}{6} = \frac{4 \pm \sqrt{7}}{3}$

The two values are $\frac{4 + \sqrt{7}}{3} \approx 2.215$ which is outside $(0,1)$ and $\frac{4 - \sqrt{7}}{3} \approx 0.451$ which is inside $(0,1)$. There seems to be an error in the options provided, since $c=2/3$ doesn't satisfy the derived equation.

Let's check if $c=2/3$ satisfies $3c^2 - 8c + 3 = 0$: $3(2/3)^2 - 8(2/3) + 3 = 3(4/9) - 16/3 + 3 = 4/3 - 16/3 + 9/3 = -3/3 = -1 \neq 0$

Since $c=2/3$ doesn't satisfy the quadratic equation, the given answer is incorrect. It should be $\frac{4 - \sqrt{7}}{3}$.

Common Mistakes & Tips

• Always verify that the value of $c$ obtained lies within the given interval $(a, b)$.
• Be careful with algebraic manipulations and the quadratic formula to avoid errors.
• Remember that polynomial functions are always continuous and differentiable.

Summary

We applied Lagrange's Mean Value Theorem to the function $f(x) = x^3 - 4x^2 + 8x + 11$ on the interval $[0, 1]$. After finding the derivative and applying the LMVT formula, we obtained a quadratic equation for $c$. Solving this equation gave us two possible values for $c$, but only one of them, $c = \frac{4 - \sqrt{7}}{3}$, lies within the interval $(0, 1)$. The provided correct answer $2/3$ is incorrect. The correct value for $c$ is $\frac{4 - \sqrt{7}}{3}$.

Final Answer

The final answer is \boxed{\frac{4 - \sqrt{7}}{3}}, which corresponds to option (D).