f(x) is a polynomial function, `f: R rarr R,` such that `f(2x)=f'(x)f''(x).` Equation f(x) = x has (A) three real and positive roots (B) three real and negative roots (C) one real root (D) three real roots such that sum of roots is zero

Options

A. three real and positive roots
B. three real and negative roots
C. one real root
D. three real roots such that sum of roots is zero

Explanation

To solve the problem, we start with the given equation:

\[ f(2x) = f'(x) f''(x) \]

### Step 1: Determine the degree of the polynomial \( f(x) \)

Let \( f(x) \) be a polynomial of degree \( n \). Then:
- The degree of \( f'(x) \) is \( n - 1 \)
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