Key Concepts and Formulas

• Rolle's Theorem: If a function $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.
• Differentiation: The power rule for differentiation states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
• Solving Systems of Equations: Techniques for solving systems of linear equations, such as substitution or elimination.

Step-by-Step Solution

Step 1: Apply Rolle's Theorem

Since Rolle's theorem holds for $f(x) = x^3 - ax^2 + bx - 4$ on $[1, 2]$, we know that $f(1) = f(2)$. This is a crucial first step because it allows us to relate the coefficients $a$ and $b$.

$f(1) = (1)^3 - a(1)^2 + b(1) - 4 = 1 - a + b - 4 = -a + b - 3$ $f(2) = (2)^3 - a(2)^2 + b(2) - 4 = 8 - 4a + 2b - 4 = -4a + 2b + 4$

Setting $f(1) = f(2)$, we get: $-a + b - 3 = -4a + 2b + 4$ $3a - b = 7 \hspace{1cm} (1)$

Step 2: Find the derivative of f(x)

We are given that $f'(\frac{4}{3}) = 0$. To use this information, we need to find the derivative of $f(x)$.

$f(x) = x^3 - ax^2 + bx - 4$ $f'(x) = 3x^2 - 2ax + b$

Step 3: Use the given information about the derivative

We know that $f'(\frac{4}{3}) = 0$, so we substitute $x = \frac{4}{3}$ into the expression for $f'(x)$:

$f'\left(\frac{4}{3}\right) = 3\left(\frac{4}{3}\right)^2 - 2a\left(\frac{4}{3}\right) + b = 0$ $3\left(\frac{16}{9}\right) - \frac{8}{3}a + b = 0$ $\frac{16}{3} - \frac{8}{3}a + b = 0$ Multiplying by 3 to eliminate fractions, we get: $16 - 8a + 3b = 0$ $8a - 3b = 16 \hspace{1cm} (2)$

Step 4: Solve the system of equations

Now we have a system of two linear equations in two variables, $a$ and $b$: $3a - b = 7 \hspace{1cm} (1)$ $8a - 3b = 16 \hspace{1cm} (2)$

We can solve this system using substitution or elimination. Let's use elimination. Multiply equation (1) by -3: $-9a + 3b = -21 \hspace{1cm} (3)$ Add equation (2) and equation (3): $(8a - 3b) + (-9a + 3b) = 16 + (-21)$ $-a = -5$ $a = 5$

Substitute $a = 5$ into equation (1): $3(5) - b = 7$ $15 - b = 7$ $b = 15 - 7$ $b = 8$

Step 5: State the ordered pair

Therefore, the ordered pair $(a, b)$ is $(5, 8)$.

Common Mistakes & Tips

• Forgetting the conditions of Rolle's Theorem: Make sure to check all three conditions (continuity, differentiability, and $f(a) = f(b)$) before applying the theorem. In this case, since we're told Rolle's Theorem holds, we can skip checking these conditions.
• Derivative Errors: Double-check your differentiation to avoid mistakes.
• Solving the System: Be careful when solving the system of equations to avoid arithmetic errors.

Summary

We applied Rolle's Theorem to the given function and interval to establish the condition $f(1)=f(2)$. This allowed us to derive a relationship between the unknown coefficients $a$ and $b$. We also used the given information about the derivative, $f'(\frac{4}{3}) = 0$, to obtain another equation relating $a$ and $b$. Solving the resulting system of two linear equations, we found the values of $a$ and $b$, which gave us the ordered pair $(5, 8)$.

Final Answer

The final answer is $\boxed{(5, 8)}$, which corresponds to option (D).