$\text{ Let }f\left(x\right)=2x^3+3x^2+6x+12\text{ and }{g}^{\text{'}}\left(x\right)=(x-1)(x-2)(x-3{)}^2\left(x^2-1\right)\left(x^2-9\right)\text{ . If }{n}_1,n_2,n_3$denote the number of local minimum, local maximum, points of inflection respectively of the function  $f\left[g\left(x\right)\right]$ then 

$\begin{array}{l}{f}^{\mathrm{\prime }}\left(x\right)=6\left(x^2+x+1\right)>0,\mathrm{\forall }x\in R\\ g^1\left(x\right)=(x+3)(x+1)(x-1{)}^2(x-2\left)\right(x-3{)}^3\\ \frac{d}{dx}\left(f\right(g\left)\right(x\left)\right)=f^{\text{'}}\left(g\right(x\left)\right)\cdot g^{\mathrm{\prime }}\left(x\right) \\ therefore f\left(g\left(x\right)\right) has local minimum at x=-1,3 \Rightarrow n_1=2\\ and has local maximum at x=-3,2 \Rightarrow n_2=2\\ and point of inflection at x=1 \Rightarrow n_3=1\\ \end{array}$