How to Use This Problem Set
Extrema and the Mean Value Theorem are closely connected. Both topics ask you to reason about what derivatives tell you about a function, but the questions require different techniques. On an AP exam, you may be asked to find an extremum, identify a critical point, determine whether a theorem applies, or explain why a particular value must exist.
Before looking at each answer, try to solve the problem completely. For theorem questions, do not stop after calculating a number: state the conditions of the theorem and explain why they are satisfied.
Extrema: identify the interval, check continuity,
find critical numbers, and compare the relevant function values.
MVT: check continuity on the closed interval and
differentiability on the open interval before calculating
.
Core Ideas to Remember
Absolute Extrema on a Closed Interval
If is continuous on the closed interval , the Extreme Value Theorem guarantees that has both an absolute maximum and an absolute minimum somewhere on that interval.
To find them, evaluate the function at every critical number in the interval and at both endpoints. Then compare the resulting values.
Critical Numbers
A critical number occurs at an interior point of the domain where the derivative is zero or does not exist. In symbols, if is defined, then a critical number can occur when:
Critical numbers are candidates for local extrema and, on a closed interval, candidates for absolute extrema. They are not automatically extrema.
Rolle's Theorem
Rolle's Theorem is a special case of the Mean Value Theorem. If is continuous on , differentiable on , and , then there is at least one number such that:
The Mean Value Theorem
If is continuous on and differentiable on , then there exists at least one such that:
Geometrically, the MVT says that somewhere between the endpoints, the instantaneous rate of change equals the average rate of change over the entire interval.
Worked Example: Absolute Extrema
Find the absolute maximum and absolute minimum of on the interval .
Worked Example: Mean Value Theorem
Verify that the Mean Value Theorem applies to on , and find all values of guaranteed by the theorem.
Practice Problems
Try each problem before opening the solution. Several problems are deliberately written in AP free-response style, where the explanation is just as important as the calculation.
Find the absolute maximum and absolute minimum of on .
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Find all critical numbers of .
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Find the absolute maximum and absolute minimum of on .
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Explain why the Extreme Value Theorem guarantees that has an absolute maximum and an absolute minimum on .
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Determine whether the Mean Value Theorem applies to on . Justify your answer.
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Verify that the Mean Value Theorem applies to on , and find the value of guaranteed by the theorem.
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Verify that Rolle's Theorem applies to on , and find all possible values of .
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A particle's position is modeled by , where is measured in seconds and is measured in meters. Explain what the Mean Value Theorem guarantees on the interval .
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A differentiable function satisfies . Determine whether has a local maximum, a local minimum, or neither at .
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Let be continuous on and differentiable on . Suppose that and . Explain why there must be a value such that .
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AP Exam Mistakes to Avoid
A critical number is only a candidate for an extremum. You still need additional information, such as a sign change in the derivative or comparison with other function values.
When finding absolute extrema on a closed interval, always evaluate the function at both endpoints as well as every critical number in the interval. The absolute maximum or minimum can occur at an endpoint.
The MVT requires both conditions: continuity on the closed interval and differentiability on the open interval. Saying only "the function is differentiable" is incomplete justification.
The MVT asks for differentiability on the open interval. If the interval is , you need differentiability on , not necessarily at the endpoints.
AP free-response questions often award a justification point before the calculation. Write the theorem conditions explicitly: "The function is continuous on the closed interval and differentiable on the open interval, so the Mean Value Theorem applies."
A Reliable Extrema Checklist
- Identify the interval.
- Check whether the function is continuous when the Extreme Value Theorem is relevant.
- Find the derivative.
- Find every critical number in the interval.
- Evaluate the function at the critical numbers and endpoints.
- Compare all values.
- State the absolute maximum and minimum clearly, including the corresponding input values.
A Reliable MVT Checklist
- Check continuity on the closed interval.
- Check differentiability on the open interval.
- Calculate the average rate of change.
- Set the derivative equal to that average rate of change.
- Solve for every value of in the open interval.
- State the result in the context of the problem.
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