Question

Let \( f(x) = ax^3 + bx^2 + cx + d \). If \( f \) has a local maximum value 21 at \( x = -1 \) and a local minimum value 7 at \( x = 1 \), then \( f(0) \) is equal to:

• A. \( 10 \)
• B. \( 11 \)
• C. \( 12 \)
• D. \( 13 \)
• E. \( 14 \)

Hint:

For cubic polynomials, setting up first derivative conditions at extrema helps find coefficients systematically.

Answer 1

Answer: (E)

Step 1: First derivative and critical points The given function is: \[ f(x) = ax^3 + bx^2 + cx + d. \] Taking the first derivative: \[ f'(x) = 3ax^2 + 2bx + c. \] Since \( x = -1 \) is a local maximum and \( x = 1 \) is a local minimum, we set \( f'(-1) = 0 \) and \( f'(1) = 0 \): \[ 3a(-1)^2 + 2b(-1) + c = 0 \] \[ 3a(1)^2 + 2b(1) + c = 0. \] Step 2: Solve for coefficients \[ 3a - 2b + c = 0 \] \[ 3a + 2b + c = 0. \] Subtracting both equations: \[ (3a + 2b + c) - (3a - 2b + c) = 0 \] \[ 4b = 0 \Rightarrow b = 0. \] Thus, the equations simplify to: \[ 3a + c = 0. \] Step 3: Using function values Since \( f(-1) = 21 \) and \( f(1) = 7 \): \[ a(-1)^3 + b(-1)^2 + c(-1) + d = 21. \] \[ a(1)^3 + b(1)^2 + c(1) + d = 7. \] \[ - a - c + d = 21 \] \[ a + c + d = 7. \] Adding both equations: \[ - a - c + d + a + c + d = 21 + 7 \] \[ 2d = 28 \Rightarrow d = 14. \]