a/ ĐKXĐ:

\(\left\{{}\begin{matrix}3-x\ge0\\x+1\ge0\\x^2-5x+6\ne0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\le3\\x\ge-1\\x\ne\left\{2;3\right\}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}-1\le x< 3\\x\ne2\end{matrix}\right.\)

b/ ĐKXĐ:

\(\left\{{}\begin{matrix}x-2m+3\ge0\\-x+m+5>0\\x\ne m\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\ge2m-3\\x< m+5\\x\ne m\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}2m-3\le x< m+5\\x\ne m\end{matrix}\right.\)

\(m+5>2m-3\Rightarrow m< 8\)

Để hàm số xác định trên \(\left(0;1\right)\)

\(\Rightarrow\left\{{}\begin{matrix}\left(0;1\right)\subset[2m-3;m+5)\\m< 8\\m\notin\left(0;1\right)\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}2m-3\le0\\m+5\ge1\\m< 8\\\left[{}\begin{matrix}m\ge1\\m\le0\end{matrix}\right.\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}-4\le m\le\frac{3}{2}\\m< 8\\\left[{}\begin{matrix}m\ge1\\m\le0\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}1\le m\le\frac{3}{2}\\-4\le m\le0\end{matrix}\right.\)

c/ ĐKXĐ: \(x\ne m\)

Để hàm số xác định trên \(\left(-1;2\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}m\le-1\\m\ge2\end{matrix}\right.\)

d/ Ta có \(a=1>0\) ; \(-\frac{b}{2a}=1\)

\(\Rightarrow\) Hàm số đồng biến trên \(\left(1;+\infty\right)\)

\(\Rightarrow\) Hàm số đồng biến trên \(\left[2;5\right]\)

\(\Rightarrow\min\limits_{\left[2;5\right]}y=y\left(2\right)=2^2-2.2+2m+3\)

\(\Rightarrow2m+3=-3\)

\(\Rightarrow m=-3\)