Glorious Gadgets is a retailer of astronomy equipment. They purchase equipment from a supplier and then sell it to customers in their store. The function $\displaystyle {C}{\left({x}\right)}={2.5}{x}+{83125}{x}^{{-{1}}}+{23750}$ models their total inventory costs (in dollars) as a function of $\displaystyle {x}$ the lot size for each of their orders from the supplier. The inventory costs include such things as purchasing, processing, shipping, and storing the equipment.

What lot size should Glorious Gadgets order to minimize their total inventory costs? (NOTE: your answer must be the whole number that corresponds to the lowest cost.)

What is their minimum total inventory cost? $