f(x) is a polynomial function, `f: R rarr R,` such that `f(2x)=f'(x)f''(x).` f(x) is (A) one-one and onto (B) one-one and into (C) many-one and onto (D) many-one and into

Options

A. one-one and onto
B. one-one and into
C. many-one and onto
D. many-one and into

Explanation

To solve the problem, we need to analyze the given equation \( f(2x) = f'(x) f''(x) \) and determine the nature of the polynomial function \( f(x) \).

### Step 1: Determine the degree of the polynomial
Let \( f(x) \) be a polynomial of degree \( n \). Then:
- The degree of \( f'(x) \) is \( n-1 \).
- The degree of \( f''(x) \) is \( n-2 \).

The left-hand side \( f(2x) \) will have a degree of \( n \) (since multiplying by 2 does not change the degree). The right-hand side \( f'(x) f''(x) \) will have a degree of \( (n-1) + (n-2) = 2n - 3 \).
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