Let  $\alpha =\sum _{k=1}^{\infty }{\sin }^{2k}\left(\frac{\pi }{6}\right)$
Let $g:[0,1]\to R$  be the function defined by  $g\left(x\right)=2^{\alpha x}+2^{\alpha (1-x)}$
Then, which of the following statements is/are NOT CORRECT?
 

 $\alpha ={\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^4+\mathrm{........}=\frac{{\left(\frac{1}{2}\right)}^2}{1-\frac{1}{4}}=\frac{1}{3}$  $\therefore g\left(x\right)=2^{x/3}+2^{(1-x)/3}=2^{x/3}+\frac{2^{1/3}}{2^{x/3}}$
 ,
Where $g\left(0\right)=1+2^{1/3}$  and  $g\left(1\right)=1+2^{1/3}$
By A.M.–G.M. inequality, we have $g\left(x\right)\ge 2\sqrt{2^{1/3}}$  and its minimum value is attained, when  $g^{/}\left(x\right)=0$   $\Rightarrow x=\frac{1}{2}$
 
 
The graph of $g^{/}\left(x\right)$  is as shown here. And  $g\left(\frac{1}{2}\right)=2^{7/6}$