Let `f(x)` be a polynomial of degree 3 such that `f(3)=1,f^(prime)(3)=-1,f^(3)=0,a n df^(3)=12.` Then the value of `f^(prime)(1)` is 12 (b) 23 (c) `-13` (d) none of these

Options

A. 12
B. 23
C. -13
D. none of these

Explanation

To solve for \( f'(1) \) given the polynomial \( f(x) \) of degree 3 with the conditions \( f(3) = 1 \), \( f'(3) = -1 \), \( f''(3) = 0 \), and \( f'''(3) = 12 \), we will follow these steps:

### Step 1: Formulate the polynomial
Since \( f(x) \) is a polynomial of degree 3, we can express it in the general form:
\[ f(x) = a(x - 3)^3 + b(x - 3)^2 + c(x - 3) + d \]

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