How do we find the inflection points of g? Consider the function $g(x)=3x(x-2)^{\frac{2}{3}}$

How do we find all the inflection points? 
I found $$g^{\prime\prime}(x)=\frac{2(5x-12)}{3(x-2)^{\frac{4}{3}}}$$ so when I do $g^{\prime\prime}(x)=0$ and when $g^{\prime\prime}$ is undefined, I get $x=12/5$ and $x=2$. Is that correct? 

After that, I replace some numbers lower/higher than $12/5$ and $2$? The thing is I did that, but it turns out I get undefined for $g^{\prime\prime}(1)$, and with $g^{\prime\prime}(2.2)$ and $g^{\prime\prime}(3)$, I get both negative numbers, so it turns out I don't have an inflexion point? Can someone please tell me what I did wrong? 

You are close, but it looks like the algebra is getting you. You are correct that the second derivative is not defined at $x = 2$. The second derivative of $g$ is certainly not undefined at $x = 1$, as the numerator is $-14$ and the denominator is $3$. Additionally, $g^{\prime\prime}(3)$ is strictly positive. This is enough to prove a concavity change of $g$ occurs as $x$ passes $2$. Try again evaluating $g^{\prime\prime}(y)$ for values of $y$ a littler larger and a little smaller than $12/5$ (but not smaller than $y= 2$). Good luck! 
