Business and Economics
60. Production level Prove that t...

Question

Business and Economics

60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a company's total cost function is given by TCFQ, which is equal to QQ minus 12Q2 plus 60Q. At what production level Q is the average cost minimized? What is the value of the average cost and marginal cost at this production level? So, the first part of this problem, we need to minimize our average cost function. So we call for the average cost function, which we'll label as AC of Q. That that is going to be equal to our total cost divided by Q. So that's going to be the quantity of Q cubed minus 12 Q squad. Plus 60Q in quantity, divided by Q. So simplifying this expression, this is going to be equal to Q2 minus 12Q plus 60. So now we need to minimize this function. And so, in order to minimize function, we first need to determine the critical points. So we'll need to calculate the derivative with respect to you. Of AC of Q. And that's gonna be equal to the derivative with respect to Q of the quantity of Q2 minus 12 Q. Plus 60 in quantity. So applying our constant multiple rule, our power rule and constant rule for derivatives. This derivative is going to be equal to 2 q minus 12. Now recall that critical points occur whenever our first derivative is either equal to zero or undefined. However, 2 q minus 12 is going to be defined everywhere, so our first derivative is also defined everywhere. So that means the only critical points are going to occur whenever a first derivative is equal to 0. So we'll set 2 Q. Minus 12 equal to 0 and solve 4 Q. So the first step is to add 12 to both sides, so we'll have 2 Q is equal to 12, and divided both sides by 2, we'll get q is equal to 6. Now we need to make sure that we do in fact have a local minimum when Q is equal to 6. So for this, we're going to use our second derivative test. So that means we're going to calculate the second derivative with respect to Q. Of a C of Q. And this is going to be equal to the derivative with respect to Q of our first derivative, which is the quantity of 2 Q minus 12. In quantity. And taking this derivative, this is just going to be equal to 2. So we see that our second derivative here is positive everywhere. So that means our second derivative is positive when Q is equal to 6. And since um our second derivative is positive when Q is equal to 6, that means that we have a local minimum. So, we do have a minimum when Q is equal to 6 for our average cost function. So that is the Q value that we were looking for. Now we need to determine the value of the average cost as well as the marginal cost at this production level when Q is equal to 6. So, the first thing we're going to do is calculate AC of 6. And this is going to be equal to. 6 squared, minus 12 multiplied by 6. Plus 60, and when we evaluate this expression, this is going to be equal to 24. So that is another part of this question answered. And so now we need to just calculate. Are marginal cost, which will label as MC of Q. And recalled that our marginal cost is equal to the derivative with respect to Q. Of our total cost, ECFQ. So that's going to be the derivative with respect of quantity of you cubed minus 12 Q squad. Plus 60Q. In quantity. And so taking this derivative, this is going to be equal to. 3 Q squad minus 24 Q plus 60. So, now we need to evaluate this expression when Q is equal to 6, so we'll need to calculate MC of 6. And that's gonna be equal to 3, multiplied by 6 squared. -24, multiplied by 6. Plus 60, and when we evaluate this expression, this is going to be equal to 24. So that is the last part of this question answered. So just to recap, we found that the average cost was minimized with a production level of Q is equal to 6. And the average cost when Q is equal to 6 is 24, and the marginal cost when Q is equal to 6 is 24 as well. So those are the answers to this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Business and Economics
60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.

Key Concepts

Average Cost

Marginal Cost

Optimization in Calculus

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