# Choose the one alternative that best completes the statement or answers the question. A...

## Question:

Choose the one alternative that best completes the statement or answers the question.

A manufacturer has determined that the total cost c of producing q units of a product is given by {eq}c= 0.04q^2+4q+6400. {/eq} The average cost will be a minimum at a production level of _____

A) 100 units

B) 200 units

C) 400 units

D) 800 units

E) none of the above

## Optimizing Functions:

Optimizing functions entail that we can find the values at which the function is at a maximum or minimum. We can do this with the help of differentiation to search for the critical points of the function. The critical point of a function can be a location where the maxima or minima is observed.

Determine the production level, q, that minimizes the average cost of the manufacturer. We determine the answer by first finding the average cost function, {eq}\displaystyle c_{ave} {/eq}, then taking its derivative, {eq}\displaystyle c'_{ave} {/eq}, equating it to zero, and solving for its root. Proceed with the solution in a step-by-step manner.

{eq}\begin{align} \displaystyle c &= 0.04q^2+4q+6400\\ \text{Take the average cost function}\\ \text{by dividing the cost function by q.}\\ c_{ave} &=\frac{0.04q^2+4q+6400}{q}\\ c_{ave} &= 0.04q+4+ \frac{6400}{q}\\ \text{Take the derivative.}\\ c'_{ave} &= \frac{d}{dq} \left(0.04q+4+ \frac{6400}{q}\right)\\ c'_{ave} &= 0.04-\frac{6400}{q^2}\\ \text{Equate it to zero.}\\ 0 &= 0.04-\frac{6400}{q^2}\\ 0.04 &= \frac{6400}{q^2}\\ q^2 &= \frac{6400}{0.04}\\ \text{Take the positive root.}\\ q &= \sqrt{ \frac{6400}{0.04}}\\ q&=\boxed{\rm 400\ units} \end{align} {/eq}

Therefore, the answer is {eq}\displaystyle \boxed{C)} {/eq}. 