If the local maximum value of the function \( f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right) \), is \( \frac{k}{e} \), then \( \left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8} \) is equal to
A) \( e^{3}+e^{6}+e^{10} \)
(B) \( e^{3}+e^{5}+e^{11} \)
(C) \( e^{3}+e^{6}+e^{11} \)
(D) \( e^{5}+e^{6}+e^{11} \)
