
Minimizing Cost
The cost of inventory depends on the ordering and storage costs according to the inventory model . Determine the order size that will minimize the cost, assuming that sales occur at a constant rate, Q is the number of units sold per year, r is the cost of storing one unit for 1 year, s is the cost of placing an order, and x is the number of units per order.
I differentiated and got:
Not sure if this is right, and not sure how to verify it. I am always supposed to explain why there is a min at x = whatever, and I can't tell if the derivative is positive or negative because of all the variables...

So you should get for the first derivative and for the second derivative. Setting the first derivative equal to zero and solving for x yields
.
We only care about the positive solution since we can't order a negative number of units. The positive solution corresponds to a local minimum, since when x is positive, the second derivative is just the product of a bunch of positive quantities (Q and s can't be negative given what they represent, and has the same sign as x) and is therefore positive. The negative solution doesn't make sense, but even if it did, it corresponds to a local maximum since the second derivative is negative if x is negative.
Quite often you can write the derivatives as the product of a bunch of quantities. If this is the case, determining the sign of the derivative is simply a matter of determining how many of those quantities are negative.
If you wanted to check all this, I'd make up values for Q, s, and r and plot the cost function. If, for example, you let , you'd expect to see a local min at and a local max at .