11. Find the maximum and minimum values of the function $y=\sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$.

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11. Find the maximum and minimum values of the function $y=\sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$.
Let $f(x)=\sqrt{x+27}+\sqrt{13-x}+\sqrt{x}, x \in[0,13]$,
then $f^{\prime}(x)=\frac{1}{2 \sqrt{x+27}}+\frac{1}{2 \sqrt{x}}-\frac{1}{2 \sqrt{13-x}}$. Since the function $y=\frac{1}{\sqrt{x+27}}+\frac{1}{\sqrt{x}}$ is monotonically decreasing on $(0,13)$, and the function $y=\frac{1}{\sqrt{13-x}}$ is monotonically increasing on $(0,13)$,
and $f^{\prime}(9)=\frac{1}{2}\left(\frac{1}{6}+\frac{1}{3}-\frac{1}{2}\right)=0 \Rightarrow f(x)$ is monotonically increasing on $(0,9)$, and monotonically
decreasing on $(9,13) \Rightarrow f(x)_{\max }=f(9)=6+2+3=11$;
$$
f(x)_{\min }=\min \{f(0), f(13)\}=\min \{3 \sqrt{3}+\sqrt{13}, 2 \sqrt{10}+\sqrt{13}\}=3 \sqrt{3}+\sqrt{13} .
$$