Finding minimum of this function I am helping my brother with his math homework and this problem has stumped me. 
I tried (150+4n)(490000/n)+0.75n as the cost function but that doesn't get me anywhere when I take second derivative.

A company needs 490,000 items per year. It costs the company \$150 to prepare a production run of these items and \$4 to produce each item. If it also costs the company $0.75 per year for each item stored, find the number of items that should be produced in each run so that total costs of production and storage are minimized.

The correct answer is 14,000 items/run. How? 

Let's say a batch is $n$ items. Then you need $750000/n$ batches, so the total yearly production cost is $450 \cdot 750000/n + 4 \cdot 750000$. Each batch lasts for $n / 750000$ years, and the average number of items on storage is $n / 2$ (if the consumption rate is constant). So the total yearly cost is:
$$
450 \cdot 750000 / n + 4 \cdot 750000 + n^2 / 1500000
$$
One would probably also take it as a given that it doesn't make sense to have a batch larger than the yearly production.
