Question

If $f\left(x\right)$ , $g\left(x\right)$ and $h\left(x\right)$ are three polynomials of degree $2$ and $\varphi \left(x\right)=$ $\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& g^{\prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\end{array}\right|$ , then ${\varphi }^{{}^{\prime }}\left(x\right)$ is

• A. a one degree polynomial
• B. a three degree polynomial
• C. a two degree polynomial
• D. a constant

Answer 1

The correct option is D a constant

Using the concept of derivative of determinants,
${\varphi }^{\prime }\left(x\right)=\left|\begin{array}{ccc}{f}^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\end{array}\right|+\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\end{array}\right|+\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime \prime }\left(x\right)& g^{\prime \prime \prime }\left(x\right)& h^{\prime \prime \prime }\left(x\right)\end{array}\right|$
$=0+0+\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ 0& 0& 0\end{array}\right|$

$=0$
Hence, ${\varphi }^{\prime }\left(x\right)$ is a constant.