Question

If Rolle's theorem holds for the function f(x) = x³ - ax² + bx - 4, x ∈ [1, 2] with f'(4/3) = 0, then ordered pair (a, b) is equal to :

• A. (5, 8)
• B. (5, -8)
• C. (-5, 8)
• D. (-5, -8)

Hint:

Rolle's theorem states that for a differentiable function where $f(a)=f(b)$, there must be at least one $c \in (a, b)$ such that $f'(c)=0$.

Answer 1

The Correct Option is A

To determine the ordered pair \( (a, b) \) for which Rolle's Theorem applies to the function \( f(x) = x^3 - ax^2 + bx - 4 \) over the interval \([1, 2]\), we follow these steps:

1. First, verify the conditions of Rolle's Theorem:
   • The function must be continuous on \([1, 2]\).
   • The function must be differentiable on \((1, 2)\).
   • The function must satisfy \( f(1) = f(2) \).
2. Calculate \( f(1) \) and \( f(2) \):
   • \( f(1) = 1^3 - a(1^2) + b(1) - 4 = 1 - a + b - 4 = b - a - 3 \)
   • \( f(2) = 2^3 - a(2^2) + b(2) - 4 = 8 - 4a + 2b - 4 = 2b - 4a + 4 \)
3. Set \( f(1) = f(2) \) to satisfy Rolle's Theorem:
   • Equating the two expressions: \( b - a - 3 = 2b - 4a + 4 \)
   • Simplify:
     • \( b - a - 2b + 4a = 4 + 3 \)
     • \( -b + 3a = 7 \)
     • This simplifies to: \( 3a - b = 7 \)     (Equation 1)
4. Next, compute \( f'(x) \) and use given \( f'(4/3) = 0 \) to find another equation:
   • Derivative: f'(x) = 3x^2 - 2ax + b
   • Given \( f'(4/3) = 0 \), substitute \( x = 4/3 \):
     • \( 3\left(\frac{4}{3}\right)^2 - 2a\left(\frac{4}{3}\right) + b = 0 \)
     • \( 3 \cdot \frac{16}{9} - \frac{8a}{3} + b = 0 \)
     • Simplifies to: \( \frac{16}{3} - \frac{8a}{3} + b = 0 \)
     • Multiply by 3 for simplicity: \( 16 - 8a + 3b = 0 \)    (Equation 2)
5. Solve the system of equations:
   • Equation 1: \( 3a - b = 7 \)
   • Equation 2: \( 16 - 8a + 3b = 0 \)
   • From Equation 1, express \( b \) in terms of \( a \): \( b = 3a - 7 \)
   • Substitute \( b = 3a - 7 \) into Equation 2:
     • \( 16 - 8a + 3(3a - 7) = 0 \)
     • \( 16 - 8a + 9a - 21 = 0 \)
     • \( a - 5 = 0 \)
     • Solve to get \( a = 5 \)
   • Substitute \( a = 5 \) back to find \( b \):
     • \( b = 3(5) - 7 = 15 - 7 = 8 \)

Thus, the ordered pair \( (a, b) \) that satisfies the conditions of Rolle's Theorem is (5, 8).