Most efficient worker time [max/min problem]

A student wrote us with the following (lightly edited):

Hello Matheno!

It’s me again. I am so grateful find you to help me about my question. I have another 2 questions that i want to ask you. I am a little bit curious and i hope that you can teach me with details on this question. Thanks Matheno!

We’ll post her other question as a separate topic.

Hi! Welcome back and thanks for the new questions.

First, let’s translate the question “At what time during the morning is the worker performing most efficiently?” into what I like to call Calculus language.

  • How are we measuring efficiency?

There are many ways to measure efficiency, but based on the function you are given, an efficient worker will likely be one who produces more units per hour, and therefore you need to find the time for which the rate of production r(t) of the worker is at its maximum.

The rate of production r(t) is just the derivative of Q(t)

r(t) = Q'(t)

and is a measure of the speed of an average worker in \, \tfrac{\text{units}}{\text{hour }} . So the question you’re trying to answer in more Calculus type language is:


What is the maximum of r(t) = Q'(t) for t\gt 0?

I’m assuming that since this is a question about finding a maximum point, you have covered maxima and minima in your course. I won’t give you the solution outright, but I will outline the steps you need to take, which you can find here on Matheno: Maxima & Minima - |

  1. Find the derivative of the function you are trying to maximize. You are trying to maximize r(t), so in this case you want to calculate the derivative r'(t).

  2. Then find all the times t>0 for which r'(t) = 0. You may know this as finding the critical points. Normally you need to look for all the places where r'(t) is undefined as well, but since Q(t) is a polynomial, all of its derivatives are well-defined everywhere.

  3. Finally, you need to show that the time(s) you found in part 2 result in either a maximum or a minimum. One way to do this is by checking the sign of r'(t) for times less than the critical point and times greater than the critical point.

Since you’re being asked for a supporting graph or diagram to help explain your solution, it might not be a bad idea to head over to, and reason about this question using a graph. Here’s my graph to get you started: Worker efficiency. We’re interested in finding the point on Q(t) where the slope of the tangent line is at its steepest (positive) slope.

As always, let me know if you have further questions about this problem, or if you want to check your solution with us!