$f\left(x\right)=\frac{x^3}{3}+\frac{x^2}{2}+x+2$ ,  $g\left(x\right)=\frac{x^4}{4}-2x^3+\frac{11}{2}{x}^2-6x$ then  $f\left(g\left(x\right)\right)$ has local minima at  $x$
 

 $f^{\text{'}}\left(x\right)=x^2+x+1>0$
 $f\left(x\right)$ is increasing then check the local minima of  $g\left(x\right)$.
 $$\begin{gathered} g^{\text{'}}\left(x\right)=x^3-6x^2+11x-6;g^{\text{'}\text{'}}\left(x\right)=3x^2-12x+11 \\ {g}^{\text{'}}\left(x\right)=(x-1)(x-2)(x-3) \\ {g}^{\text{'}\text{'}}\left(2\right)<0 \because g^{\text{'}\text{'}}\left(1\right)>0 \\ {g}^{\text{'}\text{'}}\left(3\right)>0 \end{gathered}$$
 So, local minima at  $x=1$ and   $x=3$