Short on time? This page is the recap. For the graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- Start from the model built in Topic 5.10: one function of one variable, with its domain.
- Find the critical points: where the derivative is 0 or does not exist, inside the domain.
- Justify that your answer is the absolute maximum or minimum. The method depends on the domain (table below).
- Give exactly what is asked: the location (an input), the extreme value (an output) or a quantity that follows from them.
- Interpret: name the quantity, give the value with units, and say when or where it happens.
Key relationships
| Situation | Valid justification |
|---|---|
| Closed domain [a, b] | Candidates Test: compare f at every critical point and at a and b |
| Open domain, only one critical point c | f′ changes sign at c: − to + gives the absolute minimum, + to − the absolute maximum |
| Open domain, only one critical point c | f″(c) > 0 gives the absolute minimum; f″(c) < 0 the absolute maximum |
| Any domain | f″ > 0 on the whole domain: a critical point is the absolute minimum (f″ < 0: absolute maximum) |
| The function measures… | Its maximum value means… |
|---|---|
| An amount (profit, area, volume) | the most of that amount the model allows |
| A rate (litres per minute, cars per minute) | the fastest rate, not the largest total |
| A cost, length or time to be minimised (its minimum) | the least possible cost, length or time |
Assumptions behind the method
- The function is differentiable (or at least continuous) on the domain, so its extreme values occur at critical points or endpoints.
- The domain matches the situation: lengths positive, times within the stated period, capacities respected.
- The model is only a model. The answer is the best value under the model.
Mistakes to avoid
- Stopping at f′(x) = 0 without a justification.
- Forgetting endpoints on a closed interval.
- Using f″(c) < 0 alone for an absolute maximum.
- Answering “where” when the question asks “what”, or the reverse.
- Keeping solutions outside the domain, such as a negative length.
- No units, or “it” instead of the function’s name.
- Rounding too early on calculator questions. Give final answers to three decimal places.
Quick self-check
- On [0, 10], f has one critical point at x = 4 with f(4) = 7, and f(0) = 2, f(10) = 9. What is the absolute maximum value? (9, at x = 10: the endpoint wins)
- On x > 0, g has only one critical point, at x = 5, and g″(5) > 0. What can you conclude? (g(5) is the absolute minimum of g on x > 0)
- R(t) is a rate in litres per minute, and R has its maximum at t = 12. What happens at t = 12? (The liquid flows fastest: the greatest number of litres per minute. It is not when the most liquid has flowed in total.)
Next: practice questions.