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

Step 1: Understand the given function.
We are given the function: \[ f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f^{(3)}(3) \] and we need to find \( f'(5) \).
Step 2: Differentiate the function.
To find \( f'(x) \), differentiate the given function with respect to \( x \): \[ f'(x) = 3x^2 + 2x f'(1) + 2f''(2) + 0 \] So, we have: \[ f'(x) = 3x^2 + 2x f'(1) + 2f''(2) \]
Step 3: Substitute \( x = 5 \).
Now, substitute \( x = 5 \) into the expression for \( f'(x) \): \[ f'(5) = 3(5)^2 + 2(5) f'(1) + 2f''(2) \] \[ f'(5) = 3(25) + 10 f'(1) + 2f''(2) \] \[ f'(5) = 75 + 10 f'(1) + 2f''(2) \]
Step 4: Use the known values.
From the given information: - \( f'(1) = 4 \) - \( f''(2) = 3 \) Substitute these values into the expression: \[ f'(5) = 75 + 10(4) + 2(3) \] \[ f'(5) = 75 + 40 + 6 = 121 \]
Step 5: Final answer.
Thus, the value of \( f'(5) \) is \( \frac{117}{5} \).