$x^3+y^3=3 a x y$
$3 x^2+3 y^2 \frac{d y}{d x}=3 a y+3 a x \frac{d y}{d x}$
$\begin{aligned} & \Rightarrow\left(y^2-a x\right) \frac{d y}{d x}=a y-x^2 \\ & \Rightarrow \frac{d y}{d x}=\frac{a y-x^2}{y^2-a x} \Rightarrow\left(\frac{d y}{d x}\right)\left(\frac{3 a}{2}, \frac{3 a}{2}\right)=-1 \\ & \Rightarrow\left(2 y \frac{d y}{d x}-a\right) \frac{d y}{d x}+\left(y^2-a x\right) \frac{d^2 y}{d x^2}=a \frac{d y}{d x}-2 x \\ & \Rightarrow 2 y\left(\frac{d y}{d x}\right)^2+\left(y^2-a x\right) \frac{d^2 y}{d x^2}=2 a \frac{d y}{d x}-2 x\end{aligned}$
At $\left(\frac{3 a}{2}, \frac{3 a}{2}\right) ; 2 \times \frac{3 a}{2}(-1)^2+\left(\frac{3 a^2}{4}\right)\left(\frac{d^2 y}{d x^2}\right)$
$=2 a(-1)-3 a$
$\begin{aligned} & \Rightarrow 3 a+\frac{3 a^2}{4}\left(\frac{d^2 y}{d x^2}\right)=-5 a \\ & \Rightarrow \frac{3 a}{4}\left(\frac{d^2 y}{d x^2}\right)=-8 \\ & \Rightarrow 3 a y^{\prime \prime}=-32 \\ & \Rightarrow 3 a y^{\prime \prime}+40=40-32=8\end{aligned}$