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 \(m + n + p\).