Step 1
For minimum number of sales:
\(\displaystyle\Rightarrow\ {\frac{{{d}{S}{\left({t}\right)}}}{{{\left.{d}{t}\right.}}}}={0}\)

\(\displaystyle\Rightarrow\ {\frac{{{d}{\left({5.8}{t}^{{{2}}}\ -\ {81.2}{t}\ +\ {1200}\right)}}}{{{\left.{d}{t}\right.}}}}={0}\)

\(\displaystyle\Rightarrow\ {11.6}{t}\ -\ {81.2}={0}\)

\(\displaystyle\Rightarrow\ {11.6}{t}={81.2}\)

\(\displaystyle\Rightarrow\ {t}={\frac{{{81.2}}}{{{11.6}}}}\)

\(\displaystyle\Rightarrow\ {t}={7}\) Step 2 So, minimum number of sales is given by \(\displaystyle\Rightarrow\ {S}{\left({7}\right)}={5.8}{\left({7}\right)}^{{{2}}}\ -\ {81.2}{\left({7}\right)}\ +\ {1200}\)

\(\displaystyle\Rightarrow\ {S}{\left({7}\right)}={284.2}\ -\ {568.4}\ +\ {1200}\)

\(\displaystyle\Rightarrow\ {S}{\left({7}\right)}={915.8}\)

\(\displaystyle\Rightarrow\ {S}{\left({7}\right)}\ \stackrel{\sim}{=}\ {916}\)

\(\displaystyle\Rightarrow\ {\frac{{{d}{\left({5.8}{t}^{{{2}}}\ -\ {81.2}{t}\ +\ {1200}\right)}}}{{{\left.{d}{t}\right.}}}}={0}\)

\(\displaystyle\Rightarrow\ {11.6}{t}\ -\ {81.2}={0}\)

\(\displaystyle\Rightarrow\ {11.6}{t}={81.2}\)

\(\displaystyle\Rightarrow\ {t}={\frac{{{81.2}}}{{{11.6}}}}\)

\(\displaystyle\Rightarrow\ {t}={7}\) Step 2 So, minimum number of sales is given by \(\displaystyle\Rightarrow\ {S}{\left({7}\right)}={5.8}{\left({7}\right)}^{{{2}}}\ -\ {81.2}{\left({7}\right)}\ +\ {1200}\)

\(\displaystyle\Rightarrow\ {S}{\left({7}\right)}={284.2}\ -\ {568.4}\ +\ {1200}\)

\(\displaystyle\Rightarrow\ {S}{\left({7}\right)}={915.8}\)

\(\displaystyle\Rightarrow\ {S}{\left({7}\right)}\ \stackrel{\sim}{=}\ {916}\)