1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f ( x )=|x 3 sin x cos x 6 -1 0 p p 2 p 3| where p is a constant. Then ( d 3(f( x ))/ dx 3) at x =0 is

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Q. Let $f ( x )=\begin{vmatrix}x ^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p ^2 & p ^3\end{vmatrix}$ where $p$ is a constant. Then $\frac{ d ^3(f( x ))}{ dx ^3}$ at $x =0$ is

Continuity and Differentiability

Solution:

$f ^{\prime}( x )=\begin{vmatrix}3 x ^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ p & p ^2 & p ^3\end{vmatrix} ; f ^{\prime \prime}( x )=\begin{vmatrix}6 x & -\sin x & -\cos x \\ 6 & -1 & 0 \\ p & p ^2 & p ^3\end{vmatrix}$
$f ^{\prime \prime \prime}( x )=\begin{vmatrix}6 & -\cos x & \sin x \\ 6 & -1 & 0 \\ p & p ^2 & p ^3\end{vmatrix} \Rightarrow f ^{\prime \prime \prime}(0)=\begin{vmatrix}6 & -1 & 0 \\ 6 & -1 & 0 \\ p & p ^2 & p ^3\end{vmatrix}=0$ (two identical rows) $\Rightarrow$ (D)