Question

If $f\left(x\right)=\left|\begin{array}{cc}6x^3& x^2\\ 2& x^6\end{array}\right|,$ then which of the following(s) is/are true?

• A. $f^{\prime }\left(0\right)=1$
• B. $f^{\prime }\left(0\right)=0$
• C. $f^{\prime }\left(1\right)=100$
• D. $f^{\prime }\left(1\right)=50$

Answer 1

Answer: (B), (D)

$f^{\prime }\left(1\right)=50$

$f\left(x\right)=\left|\begin{array}{cc}6x^3& x^2\\ 2& x^6\end{array}\right|$
$\Rightarrow f^{\prime }\left(x\right)=\left|\begin{array}{cc}18x^2& x^2\\ 0& x^6\end{array}\right|+\left|\begin{array}{cc}6x^3& 2x\\ 2& 6x^5\end{array}\right|$

$\Rightarrow f^{\prime }\left(1\right)=\left|\begin{array}{cc}18& 1\\ 0& 1\end{array}\right|+\left|\begin{array}{cc}6& 2\\ 2& 6\end{array}\right|\therefore f^{\prime }\left(1\right)=18+32=50$

and $f^{\prime }\left(0\right)=0$