An Inverse Inequality

Calculus Level 2

Let $f(x) = x^3 + ax^2 + bx + c$ , where $a, b$ , and $c$ are real numbers. In order for $f(x)$ to be invertible, $a$ and $b$ must be related as: $\dfrac{a^m}{b^n} \leq p$ , where $m, n$ , and $p$ are also real numbers.

Find the mimimum value of $m + n + p$ .