If a real polynomial of degree n satisfies the relation $f(x)=f'(x) f''(x)$ for all $x∈ R$. Then $f: R→ R$
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 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.