1. The vertex of this function will tell us both the maximum profit and how many units are sold to yield this profit. Using $$a=-0.01$$ and $$b=520\text{,}$$ we have:

\begin{align*} n\amp=-\frac{b}{2a}\\ \amp=-\frac{520}{2(-0.01)}\\ \amp=26000 \end{align*}

Now we will find the value of $$P$$ when $$n=26000\text{:}$$

\begin{align*} P\amp=-0.01(\substitute{26000})^2+520(\substitute{26000})-54000\\ \amp=6706000 \end{align*}

The maximum profit is $$\6{,}706{,}000\text{,}$$ which occurs if $$26{,}000$$ units are sold.

2. To find how many refrigerators need to be sold in order for the company to “break even”, we will set $$P=0$$ and solve for $$n$$ using the quadratic formula:

\begin{align*} 0\amp=-0.01n^2+520n-54000\\ n\amp=\frac{-520\pm \sqrt{520^2-4(-0.01)(-54000)}}{2(-0.01)}\\ n\amp\frac{-520\pm\sqrt{268240}}{-0.02}\\ n\amp\approx 104, n\approx 51896 \end{align*}

The company will break even if they sell about $$104$$ refrigerators or $$51{,}896$$ refrigerators.