Step 1: Understanding the Question:
The coefficients of the polynomial \( f(x) \) depend on its own derivatives evaluated at certain points. We need to find these values.
Step 2: Key Formula or Approach:
Let \( f'(1) = a \), \( f''(2) = b \), and \( f'''(3) = c \).
Then \( f(x) = 3x^3 + 2ax^2 + bx + c \).
Differentiate repeatedly to set up a system of equations for \( a, b, c \).
Step 3: Detailed Explanation:
\( f'(x) = 9x^2 + 4ax + b \)
\( f''(x) = 18x + 4a \)
\( f'''(x) = 18 \).
Thus, \( c = f'''(3) = 18 \).
From \( f''(2) = b \):
\[ b = 18(2) + 4a = 36 + 4a \]
From \( f'(1) = a \):
\[ a = 9(1)^2 + 4a(1) + b = 9 + 4a + (36 + 4a) \]
\[ a = 45 + 8a \implies -7a = 45 \implies a = -\frac{45}{7} \]
Then \( b = 36 + 4\left(-\frac{45}{7}\right) = \frac{252 - 180}{7} = \frac{72}{7} \).
The polynomial is:
\[ f(x) = 3x^3 + 2\left(-\frac{45}{7}\right)x^2 + \left(\frac{72}{7}\right)x + 18 \]
\[ f(x) = \frac{21x^3 - 90x^2 + 72x + 126}{7} \]
Step 4: Final Answer:
The function is \( \frac{1}{7}(21x^3 - 90x^2 + 72x + 126) \).