Question

If $f_1(x),f_2(x),f_3(x)$ are three polynomials of degree $2$ on $R$ and
$g(x)=\left|\begin{array}{ccc}{f}_1(x)& f_2(x)& f_3(x)\\ f_1^{\prime }(x)& f_2^{\prime }(x)& f_3^{\prime }(x)\\ f_1^{\prime \prime }(x)& f_2^{\prime \prime }(x)& f_3^{\prime \prime }(x)\end{array}\right|$,
then the value of $g(2)-g(1)$, is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

$\because g(x)=\left|\begin{array}{ccc}{f}_1(x)& f_2(x)& f_3(x)\\ f_1^{\prime }(x)& f_2^{\prime }(x)& f_3^{\prime }(x)\\ f_1^{\prime \prime }(x)& f_2^{\prime \prime }(x)& f_3^{\prime \prime }(x)\end{array}\right|$
$\Rightarrow g^{\prime }(x)=\left|\begin{array}{ccc}{f}_1^{\prime }(x)& f_2^{\prime }(x)& f_3^{\prime }(x)\\ f_1^{\prime }(x)& f_2^{\prime }(x)& f_3{}^{\prime }(x)\\ f_1^{\prime \prime }(x)& f_2^{\prime \prime }(x)& f_3^{\prime \prime }(x)\end{array}\right|+\left|\begin{array}{ccc}{f}_1(x)& f_2(x)& f_3(x)\\ f_1{}^{\prime \prime }(x)& f_2^{\prime \prime }(x)& f_3{}^{\prime \prime }(x)\\ f_1^{\prime \prime }(x)& f_2^{\prime \prime }(x)& f_3^{\prime \prime }(x)\end{array}\right|$
$+\left|\begin{array}{ccc}{f}_1(x)& f_2(x)& f_3(x)\\ f_1^{\prime }(x)& f_2^{\prime }(x)& f_3^{\prime }(x)\\ f_1^{\prime \prime \prime }(x)& f_2^{\prime \prime \prime }(x)& f_3^{\prime \prime \prime }(x)\end{array}\right|$
$=0+0+\left|\begin{array}{ccc}{f}_1(x)& f_2(x)& f_3(x)\\ f_1^{\prime }(x)& f_2^{\prime }(x)& f_3^{\prime }(x)\\ 0& 0& 0\end{array}\right|$
$=0$
$\Rightarrow g(x)=$ constant
$\therefore g(2)-g(1)=0$