In order to find the quantity to produce that minimizes the average cost, we need to first construct the average cost function. This can be done by dividing the cost function defined in this problem by the quantity.

¯C(x)=810+0.1x2x=810x+0.1xC¯(x)=810+0.1x2x=810x+0.1x

Now that we know this function, we can minimize it. The first step in the procedure to minimize a function is to find the first derivative of the function. This can be found using the Power Rule.

¯C′(x)=−810x2+0.1C¯′(x)=−810x2+0.1

Next, we need to solve for the critical points of the average cost function. This step can be completed by setting this derivative equal to zero. We will assume that x>0x>0 in our procedure, which allows us to carry out the multiplication in the second step and guarantees that we only have one solution to the resulting quadratic equation.

−810x2+0.1=00.1=810x20.1x2=810x2=8100x=90−810x2+0.1=00.1=810x20.1x2=810x2=8100x=90

To determine whether this critical point is the minimum of this function, we need to test it. The second derivative test can be completed by finding the sign of the second derivative at this value.

¯C′′(x)=1620x3¯C′′(90)=1620(90)3=−0.00222C¯″(x)=1620x3C¯″(90)=1620(90)3=−0.00222

Since the second derivative is negative when calculated at this point, this is the quantity that minimizes this function. Thus, producing 90 units will minimize the average cost.