**Answer:**

D- a minimum value of 3

**Explanation:**

\(\sqrt{-x^{2}+6 x-5}\)

Simplify/rewrite:

\[

\sqrt{-(x-5)(x-1)}

\]

Roots/zeros found at:

\[

\begin{aligned}

&x=5 \\

&x=1

\end{aligned}

\]

**First Derivative**

\begin{aligned}

&\frac{\mathrm{d}}{\mathrm{d} x}[\sqrt{-(x-5)(x-1)}]\\

&=\frac{1}{2}(-(x-5)(x-1))^{\frac{1}{2}-1} \cdot \frac{\mathrm{d}}{\mathrm{d} x}[-(x-5)(x-1)]\\

&=\frac{-\frac{\mathrm{d}}{\mathrm{d} x}[(x-5)(x-1)]}{2 \sqrt{-(x-5)(x-1)}}\\

&=-\frac{\frac{\mathrm{d}}{\mathrm{d} x}[x-5] \cdot(x-1)+(x-5) \cdot \frac{\mathrm{d}}{\mathrm{d} x}[x-1]}{2 \sqrt{-(x-5)(x-1)}}\\

&=-\frac{\left(\frac{\mathrm{d}}{\mathrm{d} x}[x]+\frac{\mathrm{d}}{\mathrm{d} x}[-5]\right)(x-1)+(x-5)\left(\frac{\mathrm{d}}{\mathrm{d} x}[x]+\frac{\mathrm{d}}{\mathrm{d} x}[-1]\right)}{2 \sqrt{-(x-5)(x-1)}}\\

&=-\frac{(1+0)(x-1)+(x-5)(1+0)}{2 \sqrt{-(x-5)(x-1)}}\\

&=-\frac{2 x-6}{2 \sqrt{-(x-5)(x-1)}}

\end{aligned}

Rewrite/simplify:

\[

=\frac{6-2 x}{2 \sqrt{-(x-5)(x-1)}}

\]

Simplify/rewrite:

\[

-\frac{x-3}{\sqrt{-(x-5)(x-1)}}

\]

Root/zero found at:

\[

x=3

\]

**Second Derivative**

\begin{array}{r}

\frac{\mathrm{d}}{\mathrm{d} x}\left[-\frac{x-3}{\sqrt{-(x-5)(x-1)}}\right] \\

=-\frac{\mathrm{d}}{\mathrm{d} x}\left[\frac{x-3}{\sqrt{-(x-5)(x-1)}}\right] \\

=-\frac{\frac{\mathrm{d}}{\mathrm{d} x}[x-3] \cdot \sqrt{-(x-5)(x-1)}-(x-3) \cdot \frac{\mathrm{d}}{\mathrm{d} x}[\sqrt{-(x-5)(x-1)}]}{(\sqrt{-(x-5)(x-1)})^{2}} \\

=\frac{\left(\frac{\mathrm{d}}{\mathrm{d} x}[x]+\frac{\mathrm{d}}{\mathrm{d} x}[-3]\right) \sqrt{-(x-5)(x-1)}-(x-3) \cdot \frac{1}{2}(-(x-5)(x-1))^{\frac{1}{2}-1} \cdot \frac{\mathrm{d}}{\mathrm{d} x}[-(x-5)(x-1)]}{(x-5)(x-1)}

\end{array}

\begin{aligned}

&=\frac{(1+0) \sqrt{-(x-5)(x-1)}-\frac{(x-3)\left(-\frac{\mathrm{d}}{\mathrm{d} x}[(x-5)(x-1)]\right)}{2 \sqrt{-(x-5)(x-1)}}}{(x-5)(x-1)}\\

&=\frac{\sqrt{-(x-5)(x-1)}+\frac{(x-3)\left(\frac{\mathrm{d}}{\mathrm{d} x}[x-5] \cdot(x-1)+(x-5) \cdot \frac{\mathrm{d}}{\mathrm{d} x}[x-1]\right)}{2 \sqrt{-(x-5)(x-1)}}}{(x-5)(x-1)}\\

&=\frac{\sqrt{-(x-5)(x-1)}+\frac{(x-3)\left(\left(\frac{\mathrm{d}}{\mathrm{d} x}[x]+\frac{\mathrm{d}}{\mathrm{d} x}[-5]\right)(x-1)+(x-5)\left(\frac{\mathrm{d}}{\mathrm{d} x}[x]+\frac{\mathrm{d}}{\mathrm{d} x}[-1]\right)\right)}{2 \sqrt{-(x-5)(x-1)}}}{(x-5)(x-1)}\\

&=\frac{\sqrt{-(x-5)(x-1)}+\frac{(x-3)((1+0)(x-1)+(x-5)(1+0))}{2 \sqrt{-(x-5)(x-1)}}}{(x-5)(x-1)}\\

&=\frac{\frac{(x-3)(2 x-6)}{2 \sqrt{-(x-5)(x-1)}}+\sqrt{-(x-5)(x-1)}}{(x-5)(x-1)}

\end{aligned}

Rewrite/simplify:

\[

=\frac{(6-2 x)(x-3)}{2(-(x-5)(x-1))^{\frac{3}{2}}}-\frac{1}{\sqrt{-(x-5)(x-1)}}

\]

Simplify/rewrite:

\[

-\frac{4}{(-(x-5)(x-1))^{\frac{3}{2}}}

\]