$\begin{aligned} & \text { } \because f(x)=x^4+a x^3+\frac{3 x^2}{2}+1 \\ & \Rightarrow f^{\prime \prime}(x)=12 x^2+6 a x+3 \\ & \because f^{\prime \prime}(x)>0 \Rightarrow 4 x^2+3 a x+1>0 \\ & \Rightarrow\left(2 x+\frac{1}{2} a\right)^2+\left(1-\frac{a^2}{4}\right)>0 \\ & \Rightarrow 1-\frac{a^2}{4}>0 \\ & \Rightarrow \frac{a^2}{4} < 1 \Rightarrow a^2 < 4 \Rightarrow-2 < a < 2 \\ & \therefore \text { Maximum value of a is } 2 .\end{aligned}$