Question

Let $f\left(x\right)$ be a polynomial with real coefficients such that $xf\left(x\right)=f^{\prime }\left(x\right)\times f^{\prime \prime }\left(x\right).$If $f\left(x\right)=0$ is satisfied $x=1,2,3$ only,then the value of $f^{\prime }\left(1\right)f^{\prime }\left(2\right)f^{\prime }\left(3\right)$is

• A. positive
• B. negative
• C. $0$
• D. Inadequate data

Answer 1

Answer: (C)

$0$

$f\left(x\right)=f^{\prime }\left(x\right)\times f^{\prime \prime }\left(x\right)$ is satisfied by only the polynomial of degree $4.$
Since, $f\left(x\right)=0$ satisfies $x=1,2,3$ only. It is clear one of the roots is twice repeated.
$\Rightarrow f^{\prime }\left(1\right)f^{\prime }\left(2\right)f^{\prime }\left(3\right)=0$