$(30$ points$)$ Suppose you have high$-$dimensional data, like megapixel images or a video with very high resolution, and you want to efficiently transfer a compressed data file consisting of all necessary features using fewer bits than the original data. Assume the efficiency, $E,$ of your data compression algorithm is modeled by

$E(n)=\frac{n^2}{10}-5n+100$

where $n$ is the number of data bits.

$($a$)$ Find the value of $n$ that maximizes efficiency using the first and second derivative tests.

$($b$)$ Interpret your results.