If a real polynomial of degree n satisfies the relation $f(x) = f'(x) f''(x) $for all $x∈ R$ Then $f: R→R$

an onto function

Let f(x) be a polynomial of degree n. Then, $f'(x)$ and $f''(x)$ are polynomials of degree $(n-1)$ and $(n-2)$ respectively.

$∴ f(x) = f'(x) f''(x) $for all $x ∈ R$

$⇒ deg (f(x)) = deg (f'(x)) + deg (f''(x))$

$⇒ n=(n-1)+n-2⇒n=3$.

Clearly, f(x), being a polynomial of degree 3, is an onto function.