Question

Let f(x) be a polynomial with real coefficients satisfies $f\left(x\right)=f^{\prime }\left(x\right)\times f{"}^{\prime }\left(x\right)$. If f(x)=0 holds good for x = 1, 2, 3 only then the value of $f^{\prime }\left(1\right)\times f^{\prime }\left(2\right)\times f^{\prime }\left(3\right)$=

• A. positive
• B. negative
• C. inadequate data

Answer 1

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