Let $f(x) = Ax^3 - 6x^2 + 5x + 12$, where $A$ is a real-valued constant. Find the value of $A$ such that $f(x)$ has a critical value at $x = -1$.
$\frac{7}{3}$
$\frac{43}{48}$
$\frac{-17}{3}$
$\frac{31}{27}$
$\frac{19}{12}$
Nothing in this list is correct.
$\frac{11}{15}$