11. (20 points) Let $x_{i} \in \mathbf{R}_{+}(i=1,2, \cdots, n, n \geqslant 2)$, $m \in \mathbf{N}, m \geqslant 2$, and $\sum_{i=1}^{n} x_{i}=T$, where $k$, $a$, $T$, and $m$ are given positive real numbers, and let $W=\sum_{i=1}^{n} \sqrt[m]{k x_{i}+a}$. Find the range of $W$.
11. Let $y_{i}=\left(k x_{i}+a\right)^{\frac{1}{m}}(i=1,2, \cdots, n)$. Then
$$
y_{i}^{m}=k x_{i}+a \Rightarrow \sum_{i=1}^{n} y_{i}^{m}=k T+n a \text {. }
$$

By Jensen's inequality, we have
$$
\begin{array}{l}
\frac{\sum_{i=1}^{n} y_{i}^{m}}{n} \geqslant\left(\frac{\sum_{i=1}^{n} y_{i}}{n}\right)^{m}=\left(\frac{W}{n}\right)^{m} \\
\Rightarrow\left(\frac{k T+n a}{n}\right)^{\frac{1}{m}} \geqslant \frac{W}{n} \Rightarrow W \leqslant n \sqrt[m]{\frac{k T+n a}{n}} \text {. } \\
\text { Also, } \frac{\left(k x_{i}+a\right)^{\frac{1}{m}}-a^{\frac{1}{m}}}{1} \\
=\frac{\left(k x_{i}+a\right)-a}{\left(k x_{i}+a\right)^{\frac{m-1}{m}}+\left(k x_{i}+a\right)^{\frac{m-2}{m}} a^{\frac{1}{m}}+\cdots+a^{\frac{m-1}{m}}} \\
k x_{i} \\
=\frac{k x_{i}}{\left(k x_{i}+a\right)^{\frac{m-1}{m}}+\left(k x_{i}+a\right)^{\frac{m-2}{m}} a^{\frac{1}{m}}+\cdots+a^{\frac{m-1}{m}}} \text {. } \\
\text { Let } \lambda=\left(k x_{i}+a\right)^{\frac{m-1}{m}}+\left(k x_{i}+a\right)^{\frac{m-2}{m}} a^{\frac{1}{m}}+\cdots+a^{\frac{m-1}{m}} \text {. } \\
\text { Since } 0\frac{k x_{i}\left[(k T+a)^{\frac{1}{m}}-a^{\frac{1}{m}}\right]}{(k T+a)-a}$

Combining with equation (1), we get
$$
\begin{array}{l}
\left(k x_{i}+a\right)^{\frac{1}{m}}>\frac{1}{T}\left[(k T+a)^{\frac{1}{m}}-a^{\frac{1}{m}}\right] x_{i}+a^{\frac{1}{m}} \\
\Rightarrow W>\frac{1}{T}\left[(k T+a)^{\frac{1}{m}}-a^{\frac{1}{n}}\right]\left(x_{1}+x_{2}+\cdots+x_{n}\right)+n a^{\frac{1}{m}} \\
\quad=\sqrt[m]{k T+a}+(n-1) \sqrt[m]{a} .
\end{array}
$$

Therefore, the range of $W$ is
$$
\left[\sqrt[m]{k T+a}+(n-1) \sqrt[m]{a}, n \sqrt[m]{\frac{k T+n a}{n}}\right] .
$$

When $x_{1}=x_{2}=\cdots=x_{n}=\frac{T}{n}$, $W=n \sqrt[m]{\frac{k T+n a}{n}}$;
When $x_{1}, x_{2}, \cdots, x_{n-1} \rightarrow 0, x_{n} \rightarrow T$,
$$
W \rightarrow(n-1) \sqrt[m]{a}+\sqrt[m]{k T+a} .
$$