Question

Let \( f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \), \( x \in \mathbb{R} \). Then \( f'(10) \) is equal to ______.

Answer 1

Correct Answer: 202

Step 1. Given \( f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \). Substitute \( f'(1) = -5 \), \( f''(2) = 2 \), \( f'''(3) = 6 \).

Step 2. Calculate \( f'(x) \):  
  \( f'(x) = 3x^2 + 2x f'(1) + f''(2) \)

Step 3. Evaluate \( f'(10) \):  
  \( f'(10) = 202 \)

Answer 2

Given function:

\[ f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \]

Step 1: Differentiate \( f(x) \)

\[ f'(x) = 3x^2 + 2x f'(1) + f''(2) \]

Step 2: Second derivative

\[ f''(x) = 6x + 2f'(1) \]

Step 3: Third derivative

\[ f'''(x) = 6 \]

Step 4: Substitute given values

\[ f'(1) = -5, \quad f''(2) = 2, \quad f'''(3) = 6 \] So, \[ f(x) = x^3 + x^2(-5) + x(2) + 6 = x^3 - 5x^2 + 2x + 6 \]

Step 5: Find \( f'(x) \)

\[ f'(x) = 3x^2 - 10x + 2 \]

Step 6: Evaluate \( f'(10) \)

\[ f'(10) = 3(10)^2 - 10(10) + 2 = 300 - 100 + 2 = 202 \] 

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Final Answer:

\[ \boxed{202} \]