Which of the following function/functions has/ have point of inflection ?

$f\left(x\right)=x^{6/7} \text{or }{f}^{\prime \prime }\left(x\right)=-\frac{6}{7}\frac{13}{7}{x}^{\frac{-8}{7}}$

Here, $f"\left(x\right)$  does not change sign. Hence, it has no point of inflection.

For $f\left(x\right)=x^6\Rightarrow f^{\mathrm{\prime }}\left(x\right)=30x^4\Rightarrow f\text{'}\text{'}\left(x\right)=120x^3$ , but   $f"\left(x\right)$ does not change sign in the neighborhood of x=0

$f\left(x\right)=\cos x+2x \Rightarrow f\text{'}\left(x\right)=-\sin x+2\Rightarrow f^{\prime \prime }\left(x\right)=-\cos x$  or

$f^{\prime \prime }\left(0\right)=0 \text{for }x=(2n+1)\pi /2,n\in Z$

Also , sign of $f"\left(x\right)$  changes sign in the neighborhood of $x=(2n+1)\pi /2$ . Hence, function has infinite points of inflection. 

$f\left(x\right)=x\left|x\right|=\{\begin{array}{cc}-x^2,& x<0\\ x^2,& x\ge 0\end{array}\Rightarrow f^{\prime \prime }(x)=\{\begin{array}{cc}-2,& x<0\\ 2,& x>0\end{array}$

Here, $f"\left(x\right)$  changes sign in the neighborhood of $x=0$ . Hence has point of inflection.