One thing you can do is recognize that at the minimum value, $x$ , the function must be locally convex, so that a small perturbation $t$ to the left and right of $x$ should result in the same value as $t \rightarrow 0$ .
Therefore, we can say that as $t \rightarrow 0$ , it must be that $$\frac{(x+t)^3}{x+t-6} + \frac{(x-t)^3}{x-t-6} = 2\frac{(x+t)^3}{x+t-6}.$$
If we solve this equation for $t$ in terms of $x$ , we find that
$$t = \pm x\sqrt{\frac{x-9}{x+3}}.$$
So, for this equation to hold as $t \rightarrow 0$ , we can see that $x$ must be equal to 9.