Question

Let \( f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f^{(3)}(3) \) for all \( x \in \mathbb{R} \). Then the value of \( f'(5) \) is:

• A. \( \frac{109}{5} \)
• B. \( \frac{117}{5} \)
• C. \( \frac{119}{5} \)
• D. \( \frac{118}{5} \)

Hint:

When differentiating a function involving other functions and constants, be careful to apply the rules of differentiation correctly and substitute any given values at the right steps.

Answer 1

The Correct Option is B

Given:
\[ f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f^{(3)}(3) \] for all \(x \in \mathbb{R}\).

Step 1: Find derivatives of \(f(x)\)

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

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

Third derivative: \[ f^{(3)}(x) = 6 \]

Step 2: Use the given functional values

Since \(f^{(3)}(x) = 6\) for all \(x\),

\[ f^{(3)}(3) = 6 \]

From second derivative:

\[ f''(2) = 6(2) + 2f'(1) = 12 + 2f'(1) \]

Step 3: Find \(f'(1)\)

Substitute \(x = 1\) in \(f'(x)\):

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

\[ f'(1) = 3 + 2f'(1) + 2(12 + 2f'(1)) \]

\[ f'(1) = 3 + 2f'(1) + 24 + 4f'(1) \]

\[ f'(1) = 27 + 6f'(1) \]

\[ -5f'(1) = 27 \Rightarrow f'(1) = -\frac{27}{5} \]

Step 4: Find \(f'(5)\)

First find \(f''(2)\):

\[ f''(2) = 12 + 2\left(-\frac{27}{5}\right) = \frac{60 - 54}{5} = \frac{6}{5} \]

Substitute in \(f'(x)\):

\[ f'(x) = 3x^2 + 2x\left(-\frac{27}{5}\right) + 2\left(\frac{6}{5}\right) \]

\[ f'(x) = 3x^2 - \frac{54x}{5} + \frac{12}{5} \]

Put \(x = 5\):

\[ f'(5) = 3(25) - \frac{54(5)}{5} + \frac{12}{5} \]

\[ f'(5) = 75 - 54 + \frac{12}{5} = 21 + \frac{12}{5} = \frac{117}{5} \]

Final Answer:

\[ \boxed{f'(5) = \frac{117}{5}} \]