11. Given that $a, b$ are positive constants, and $x$ is an acute angle, find the minimum value of the function $y=a \sin ^{n} x+b \cos ^{n} x$.
11.
$$\begin{array}{c}
\left(a \sin ^{n} x+b \cos ^{n} x\right)^{\frac{2}{n}}\left(a^{-\frac{2}{n-2}}+b^{-\frac{2}{n-2}}\right)^{\frac{n-2}{n}} \geqslant \\
a^{\frac{2}{n}} \sin ^{2} x\left(a^{-\frac{2}{n-2}}\right)^{\frac{n-2}{n}}+b^{\frac{2}{n}} \cos ^{2} x\left(b^{-\frac{2}{n-2}}\right)^{\frac{n-2}{n}}=1 \\
y^{2}\left(a^{-\frac{2}{n-2}}+b^{-\frac{2}{n-2}}\right)^{\frac{n-2}{n}} \geqslant 1
\end{array}$$

Since $y>0$, we have $y \geqslant\left(a^{-\frac{2}{n-2}}+b^{-\frac{2}{n-2}}\right)^{-\frac{n-2}{2}}$. Therefore, the minimum value of $y$ is
$$\left(a^{-\frac{2}{n-2}}+b^{-\frac{2}{n-2}}\right)^{-\frac{n-2}{2}}$$