Let $f(x)=\begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$ where $q$ is any constant, then what is $\dfrac{d^2}{dx^2}\big(f(x)\big)$ at $x=0$ equal to ?

Given the function $f(x)$ as a determinant where only the first row contains the variable $x$:

$f(x)=\begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$

To differentiate the determinant with respect to $x$, we differentiate the elements of the first row while keeping the other rows unchanged.

Differentiating once with respect to $x$:

$f'(x)=\begin{vmatrix} 6x & -\sin x & -\cos x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$

Differentiating again with respect to $x$:

$f''(x)=\begin{vmatrix} 6 & -\cos x & \sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$

Substituting $x=0$ into the second derivative:

$f''(0)=\begin{vmatrix} 6 & -1 & 0 \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$

Since the first and second rows of the determinant are identical, the value of the determinant is zero.

$f''(0) = 0$

Answer: $0$