If $\mathrm{a}\in (0,1),\text{ and }\mathrm{f}\left(\mathrm{a}\right)=\left({\mathrm{a}}^2-\mathrm{a}+1\right)+\frac{8{\sin }^2\mathrm{a}}{\sqrt{{\mathrm{a}}^2-\mathrm{a}+1}}+\frac{27{\mathrm{cosec}}^2\mathrm{a}}{\sqrt{{\mathrm{a}}^2-\mathrm{a}+1}}$, then the least value of f(a) is

 Using A.M. $\ge$ G.M. (for positive numbers) 

$\left({\mathrm{a}}^2-\mathrm{a}+1\right)+\frac{8{\sin }^2\mathrm{a}}{\sqrt{{\mathrm{a}}^2-\mathrm{a}+1}}+\frac{27{\mathrm{cosec}}^2\mathrm{a}}{\sqrt{{\mathrm{a}}^2-\mathrm{a}+1}}$

$\begin{array}{l}\ge 3{\left(\left({\mathrm{a}}^2-\mathrm{a}+1\right)\frac{8{\sin }^2\mathrm{a}}{\sqrt{\left({\mathrm{a}}^2-\mathrm{a}+1\right)}}\frac{27{\mathrm{cosec}}^2\mathrm{a}}{\sqrt{\left({\mathrm{a}}^2-\mathrm{a}+1\right)}}\right)}^{\frac{1}{3}}\\ =3{\left(2^3{3}^3\right)}^{\frac{1}{3}}=3\left(6\right)=18\end{array}$