Minimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?

Find the minimum of $y=f(x)=\dfrac{x^3}{x-6}$ for $x>6$.

I can solve the question using derivatives but I have no any idea how to do it without them. Using derivatives, we find $x=9$ and $y_{min}=243$.

When looking for the minimum of a function, calculus is the default choice (with good reason), but it is a relatively new idea in Mathematics. Questions such as the one posed here are an interesting intellectual challenge because the obvious approach (calculus) is not the only approach. Without calculus there is no obvious approach though...