here a function and its derivatives accordingly by: $$f(x)=x-6x^{1/3}$$ $$f'(x)=1-2x^{-2/3}$$ $$f''(x)=\frac{4}{3}x^{-5/3}$$
as we can't get zero for $f''(x)$ and when i try to make derivative not exist on $f''(x)$ it would also make $f'(x)$ not exist at same time. Which should leave us no other choice than observe the concavity directly from the original function , but exactly how to solve this?
An inflection point occurs when the value of $f''(x)$ changes sign and the curve $f(x)$ is continuous. In our case $f''(x)\gt0$ for all $x\gt0$ and $f''(x)\lt0$ for all $x\lt0$. Although $f''(0)$ is undefined, the value of $f''(x)$ changes sign at the point where $x=0$ and $f(x)$ is continuous at this point hence $(0,0)$ is the only inflection point of $f(x)$.