I sketched the function $f(x) = x^{6/7}-9x^{2/7}$ and got something like this.
Where POI means point of inflection.
However, when I graph it in Desmos, I get what looks like an oblique asymptote, that corresponds to an absolute value function.
The more I zoom out though, the more the slope seems to decrease.
This shows (or at least lends credence) to the fact that there are two POIs, right? In addition, is it true that there is no asymptote?
This is my solution.
In fact, there is no contradiction concerning your POI, with abscissas at $\pm (15)^{7/4} \approx \pm 114.3$ : it is impossible to spot them even on an large curve plainly because the transition from positive to negative concavity is very faint.
See graphics below obtained with Geogebra. The first one for the variations of function $f$, the second one for function $f''$, the latter graphics evidencing an extremely small variation (order $10^{-5}$), before and after the abscissa the transition at the POI.
Remark: $f''(x)=-\dfrac{6}{49}\dfrac{x^{4/7}-15}{x^{12/7}}.$
$$\text{Above: Curve of f}.$$
$$\text{Above: Curve of} \ f'' \ \text{ in the vicinity of a POI}.$$