Question

Let \( f(x) = ax^3 + bx^2 + ex + 41 \) be such that \( f(1) = 40 \), \( f'(1) = 2 \) and \( f''(1) = 4 \). Then \( a^2 + b^2 + c^2 \) is equal to:

• A. 60
• B. 73
• C. 54
• D. 51

Answer 1

The Correct Option is D

Step 1: Derivatives of \(f(x)\) The given function is:

\(f(x) = ax^3 + bx^2 + cx + 41.\)

The first derivative:

\(f'(x) = 3ax^2 + 2bx + c.\)

The second derivative:

\(f''(x) = 6ax + 2b.\)

Step 2: Use the given conditions

1. From \(f'(1) = 2\):
2. From \(f''(1) = 4\):
3. From \(f(1) = 40\):

Step 3: Solve for \(a\), \(b\), \(c\)

From equation (2):

\(6a + 2b = 4 \implies 3a + b = 2.\) (4)

From equations (1) and (4):

\(3a + 2b + c = 2,\)

\(3a + b = 2.\)

Subtract equation (4) from (1):

\((3a + 2b + c) - (3a + b) = 2 - 2,\)

\(b + c = 0.\) (5)

From equations (3) and (5):

\(a + b + c = -1, \quad b + c = 0.\)

Subtract:

\(a = -1.\) (6)

Substitute \(a = -1\) into equation (4):

\(3(-1) + b = 2 \implies -3 + b = 2 \implies b = 5.\) (7)

From equation (5):

\(b + c = 0 \implies 5 + c = 0 \implies c = -5.\) (8)

Step 4: Compute \(a^2 + b^2 + c^2\)

\(a^2 + b^2 + c^2 = (-1)^2 + 5^2 + (-5)^2 = 1 + 25 + 25 = 51.\)

Final Answer: Option (4).