$\ln y=\sin ^2 x \cdot \ln \left(\frac{\sqrt{3 e}}{2 \sin x}\right)$
$\frac{1}{y} y^{\prime}=\ln \left(\frac{\sqrt{3 e}}{2 \sin x}\right) 2 \sin x \cos x+\sin ^2 x \frac{2 \sin x}{\sqrt{3 e}} \frac{\sqrt{3 e}}{2}(-\operatorname{cosec} x \cot x)$
$\frac{ dy }{ dx }=0 \Rightarrow \ln \left(\frac{\sqrt{3 e }}{2 \sin x }\right) 2 \sin x \cos x -\sin x \cos x =0$
$\Rightarrow \sin x \cos x\left[2 \ln \left(\frac{\sqrt{3 e }}{2 \sin x }\right)-1\right]=0$
$\Rightarrow \ln \left(\frac{3 e }{4 \sin ^2 x }\right)=1 \Rightarrow \frac{3 e }{4 \sin ^2 x }= e \Rightarrow \sin ^2 x =\frac{3}{4}$
$\Rightarrow \sin x =\frac{\sqrt{3}}{2} \quad\left(\text { as } x \in\left(0, \frac{\pi}{2}\right)\right)$
$\Rightarrow \text { local max value }=\left(\frac{\sqrt{3 e }}{\sqrt{3}}\right)^{3 / 4}= e ^{3 / 8}=\frac{ k }{ e }$
$\Rightarrow k ^8= e ^{11}$
$\Rightarrow\left(\frac{ k }{ e }\right)^8+\frac{ k ^8}{ e ^5}+ k ^8= e ^3+ e ^6+ e ^{11}$
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and $det(A) = det(4I)$, where $I$ is $3 × 3$ identity matrix, then $(a -b)^3 + (b -c)^3 + (c -a)^3$ can be equal to -