Question

A non-zero polynomial with real coefficients has the property that $f\left(x\right)=f^{\prime }\left(x\right)\cdot f^{\prime \prime }\left(x\right).$ The leading coefficient of $f\left(x\right)$ is

• A. $\frac{1}{6}$
• B. $\frac{1}{9}$
• C. $\frac{1}{12}$
• D. $\frac{1}{18}$

Answer 1

The correct option is D $\frac{1}{18}$

Let degree of $f\left(x\right)$ be $n.$
Then, degree of $f^{\prime }\left(x\right)=n-1$
and degree of $f^{\prime \prime }\left(x\right)=n-2$

Hence $n=(n-1)+(n-2)$
$\Rightarrow n=2n-3$
$\Rightarrow n=3$

Now, let $f\left(x\right)=a{x}^3+b{x}^2+cx+d,$ where $a\ne 0$
$f^{\prime }\left(x\right)=3a{x}^2+2bx+c$
$f^{\prime \prime }\left(x\right)=6ax+2b$

$\therefore a{x}^3+b{x}^2+cx+d=(3a{x}^2+2bx+c)(6ax+2b)$
Comparing the coefficient of $x^3,$
$18a^2=a$
$\Rightarrow a=\frac{1}{18}$