Science:Math Exam Resources/Courses/MATH104/December 2016/Question 06 (a)

MATH104 December 2016

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Question 06 (a)
Suppose that a company's Cost function is an increasing and differentiable function. Use calculus to show that average cost will be minimized at the level of production when the average cost is equal to the marginal cost.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall that if the cost is $C(q)$ as a function of the quantity $q$ , then the average cost is ${\frac {C(q)}{q}}$ and the marginal cost is $C'(q)={\frac {dC}{dq}}(q)$ .

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Following the hint, we minimize the average cost $C_{\text{avg}}(q)={\frac {C(q)}{q}}$ by finding its critical points. The derivative of the average cost (with respect to quantity) is $C_{\text{avg}}'(q)={\frac {C'(q)q-C(q)}{q^{2}}}$ by the quotient rule, which vanishes exactly when the numerator is zero. In other words, the critical points of $C_{\text{avg}}$ satisfy $C'(q)={\frac {C(q)}{q}},$ which by definition means that the average cost is equal to the marginal cost.