• 62 questions

AP Calculus Unit 5 Practice Test

Unofficial practice for Unit 5 of AP Calculus AB and BC: Mean Value Theorem, extrema, concavity, curve sketching and optimization.

Which conditions on f let the Mean Value Theorem be used on the interval from a to b?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 14, 2026.

Next step AP Calculus Unit 4: related rates and motionAP Calculus integrals: Unit 6

All 62 questions

Topics 5.1 and 5.2: Mean value theorem, extreme value theorem, critical points 12

  • A differentiable function has f(1) = 2 and f(5) = 14. What value must f′(c) take for some c between 1 and 5? Wikipedia, "Mean value theorem"; course topic 5.1
  • A driver covers 150 miles of highway in 2.5 hours. What does the Mean Value Theorem guarantee about the speedometer? Wikipedia, "Mean value theorem"; course topic 5.1
  • f(x) = 1/x has no maximum value on the interval (0, 1]. Why does this not contradict the Extreme Value Theorem? Wikipedia, "Extreme value theorem"; course topic 5.2
  • f(x) = x³ has f′(0) = 0. What does f have at x = 0? Wikipedia, "Critical point (mathematics)"; course topic 5.2
  • For f(x) = x² − 4x on [0, 4], f(0) = f(4) = 0. At which c in (0, 4) is f′(c) = 0? Wikipedia, "Mean value theorem"; course topic 5.1
  • For f(x) = x³ − x on the interval [0, 2], at which c does f′(c) equal the average rate of change of f? Wikipedia, "Mean value theorem"; course topic 5.1
  • What are the critical points of f(x) = x⁴ − 8x²? Wikipedia, "Critical point (mathematics)"; course topic 5.2
  • What is the only critical point of f(x) = xe^(2x)? Wikipedia, "Critical point (mathematics)"; course topic 5.2
  • Where is the critical point of f(x) = x^(2/3), and why is it one? Wikipedia, "Critical point (mathematics)"; course topic 5.2
  • Which condition guarantees, by the Extreme Value Theorem, that f has both an absolute maximum and an absolute minimum on an interval? Wikipedia, "Extreme value theorem"; course topic 5.2
  • Which conditions on f let the Mean Value Theorem be used on the interval from a to b? Wikipedia, "Mean value theorem"; course topic 5.1
  • Why can the Mean Value Theorem not be used for f(x) = |x − 1| on the interval [0, 3]? Wikipedia, "Mean value theorem"; course topic 5.1
Show all 62 questions (50 more)

Topics 5.3 and 5.4: Increasing, decreasing and the first derivative test 10

  • A function f has derivative f′(x) = (x − 1)²(x + 2). On which interval is f increasing? Wikipedia, "Monotonic function"; course topic 5.3
  • A function f has derivative f′(x) = x(x − 3)²(x + 2). At which value of x does f have a local minimum? Wikipedia, "Derivative test"; course topic 5.4
  • f(x) = x²e^x has two critical points. At which one is there a local maximum? Wikipedia, "Derivative test"; course topic 5.4
  • For f(x) = 3x^(1/3) − x, the derivative f′(x) = x^(−2/3) − 1 does not exist at x = 0. What does f have at x = 0? Wikipedia, "Derivative test"; course topic 5.4
  • The graph of f′ is the line through (0, −6) and (3, 0). On which interval is f decreasing? Wikipedia, "Monotonic function"; course topic 5.3
  • On which interval is f(x) = 2x³ + 3x² − 36x decreasing? Wikipedia, "Monotonic function"; course topic 5.3
  • On which interval is f(x) = e^x − 3x increasing? Wikipedia, "Monotonic function"; course topic 5.3
  • On which interval is f(x) = ln x − x increasing? Wikipedia, "Monotonic function"; course topic 5.3
  • What local extrema does f(x) = x⁴ − 4x³ have? Wikipedia, "Derivative test"; course topic 5.4
  • Which statement is a correct justification that f has a local minimum at x = 2? Wikipedia, "Derivative test"; course topic 5.4

Topic 5.5: The candidates test 6

  • At which x does f(x) = x + 2 cos x reach its absolute maximum on [0, π]? Wikipedia, "Maximum and minimum"; course topic 5.5
  • f is continuous on [0, 6] and its only critical points are x = 2 and x = 5. If f(0) = 3, f(2) = 7, f(5) = −1 and f(6) = 4, what is the absolute minimum value of f on [0, 6]? Wikipedia, "Maximum and minimum"; course topic 5.5
  • For a continuous function on [a, b], which x-values are the candidates for the absolute extrema? Wikipedia, "Maximum and minimum"; course topic 5.5
  • What is the absolute maximum value of f(x) = x² − 4x + 1 on the interval [0, 5]? Wikipedia, "Maximum and minimum"; course topic 5.5
  • What is the absolute minimum value of f(x) = xe^(−x/2) on the interval [0, 6]? Wikipedia, "Maximum and minimum"; course topic 5.5
  • What is the absolute minimum value of f(x) = x³ − 6x² + 5 on the interval [−1, 5]? Wikipedia, "Maximum and minimum"; course topic 5.5

Topics 5.6 and 5.7: Concavity and the second derivative test 11

  • A continuous function on an interval has exactly one critical point, at x = c, and it is a local minimum. What follows? Wikipedia, "Derivative test"; course topic 5.7
  • A function f has second derivative f″(x) = x²(x − 4). Where does the graph of f have a point of inflection? Wikipedia, "Inflection point"; course topic 5.6
  • A twice-differentiable function has f′(0) = 5, f′(1) = 3, f′(2) = 2 and f′(3) = 1.5, and f′ keeps decreasing between those points. On [0, 3] the graph of f is: Wikipedia, "Second derivative"; course topic 5.6
  • A twice-differentiable function has f′(1) = 0, f″(1) = −4, f′(6) = 0 and f″(6) = 0. What does the second derivative test conclude? Wikipedia, "Derivative test"; course topic 5.7
  • At which x-values does the graph of f(x) = x⁴ − 6x² have points of inflection? Wikipedia, "Inflection point"; course topic 5.6
  • f(x) = sin x + cos x has f′(π/4) = 0. What does the second derivative test show at x = π/4? Wikipedia, "Derivative test"; course topic 5.7
  • f(x) = x ln x, for x > 0, has one critical point, at x = 1/e. What does the second derivative test show there? Wikipedia, "Derivative test"; course topic 5.7
  • f(x) = x³ − 3x² − 24x has a critical point at x = −2. What does the second derivative test show there? Wikipedia, "Derivative test"; course topic 5.7
  • On an interval where the graph of f is concave down, what is true of f′? Wikipedia, "Second derivative"; course topic 5.6
  • On which interval is the graph of f(x) = ln(x² + 1) concave up? Wikipedia, "Second derivative"; course topic 5.6
  • On which interval is the graph of f(x) = x³ − 9x² + 4 concave up? Wikipedia, "Second derivative"; course topic 5.6

Topics 5.8 and 5.9: Connecting f, f′ and f″ 8

  • A cubic polynomial has a local maximum at x = −1 and a local minimum at x = 3. Where is its point of inflection? Wikipedia, "Inflection point"; course topic 5.9
  • A function has derivative f′(x) = x² − 4x + 3. Where does the graph of f have a point of inflection? Wikipedia, "Inflection point"; course topic 5.9
  • A twice-differentiable function has f(2) = 3, f′(2) = 0 and f″(2) = −1. What does its graph look like near x = 2? Wikipedia, "Second derivative"; course topic 5.8
  • f″(x) > 0 for every x in the interval (0, 4). What must be true of f′ on that interval? Wikipedia, "Second derivative"; course topic 5.8
  • The graph of f is a parabola that opens downward with its vertex at x = 1. What is the graph of f′? Wikipedia, "Second derivative"; course topic 5.8
  • The graph of f′ crosses the x-axis from above to below at x = 2. What does the graph of f have at x = 2? Wikipedia, "Second derivative"; course topic 5.8
  • The graph of f′ has a local maximum at x = 1. What does the graph of f have at x = 1? Wikipedia, "Inflection point"; course topic 5.9
  • On an interval the graph of f is rising and concave down. What are the signs of f′ and f″ there? Wikipedia, "Second derivative"; course topic 5.8

Topics 5.10 and 5.11: Optimization 9

  • A closed cylindrical can must hold 16π cubic centimeters. Which radius uses the least metal? Wikipedia, "Maximum and minimum"; course topic 5.10
  • A rectangle has its base on the x-axis and its two upper corners on the parabola y = 12 − x². If the right-hand upper corner is at (x, y), which function gives the area? Wikipedia, "Maximum and minimum"; course topic 5.10
  • A rectangle has its base on the x-axis and its upper corners on y = 12 − x². What is the largest area it can have? Wikipedia, "Maximum and minimum"; course topic 5.10
  • A rectangular pen is built against a straight river, so only three sides need fence. With 120 meters of fence, what is the largest area the pen can have? Wikipedia, "Maximum and minimum"; course topic 5.10
  • A shop's weekly profit from selling x units is P(x) = −2x² + 80x − 300 dollars. How many units give the largest profit? Wikipedia, "Maximum and minimum"; course topic 5.10
  • A tray with no lid is folded from a card 18 centimeters square after a square of side x is cut from each corner. Which x gives the tray the largest volume? Wikipedia, "Maximum and minimum"; course topic 5.10
  • C(v) is a truck's fuel cost in dollars per kilometer at a speed of v kilometers per hour. C has its absolute minimum at v = 80, with C(80) = 0.12. What does this mean? Wikipedia, "Maximum and minimum"; course topic 5.11
  • Two positive numbers have a product of 50. What is the smallest possible value of the first number plus twice the second? Wikipedia, "Maximum and minimum"; course topic 5.10
  • Which point on the line y = 2x + 5 is closest to the origin? Wikipedia, "Maximum and minimum"; course topic 5.10

Topic 5.12: Implicit relations 6

  • A curve has dy/dx = (3x² − 2)/(2y − 6). Where can its tangent line be vertical? Wikipedia, "Implicit function"; course topic 5.12
  • At the point (0, 2) on the ellipse x² + 4y² = 16, dy/dx = 0 and d²y/dx² = −1/8. What does the curve have at that point? Wikipedia, "Implicit function"; course topic 5.12
  • At which point does the curve y² − 4y = x have a vertical tangent line? Wikipedia, "Implicit function"; course topic 5.12
  • For the curve x² + xy + y² = 12, dy/dx = −(2x + y)/(x + 2y). At which points is the tangent line horizontal? Wikipedia, "Implicit function"; course topic 5.12
  • For the hyperbola x² − y² = 7, dy/dx = x/y. What is d²y/dx² in terms of y? Wikipedia, "Implicit function"; course topic 5.12
  • The point (2, 2) lies on the curve y³ − xy = 4. Is y increasing or decreasing as x increases through that point? Wikipedia, "Implicit function"; course topic 5.12

How this list was compiled: Our own questions, written to Unit 5 (Analytical Applications of Differentiation; topics 5.1 to 5.12) of the College Board's AP Calculus AB and BC Course and Exam Description, https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf, read 2026-10-14. Every answer is mathematics worked from what the question prints, and the computed ones were worked out a second time with computer algebra on 2026-10-14. The rule, theorem or definition behind each answer was read on 2026-10-14 in the Wikipedia article named under the question. Licence: Own work.

How it works

This AP Calculus Unit 5 practice test has 62 questions of our own on Analytical Applications of Differentiation. It is unofficial: this site is not affiliated with or endorsed by the College Board, the owner of AP.

Read the full guide

What Unit 5 covers

The College Board's course description, read October 14, 2026, gives Unit 5 twelve topics, from the Mean Value Theorem (5.1) to implicit relations (5.12). Seven sets follow them in order: the two existence theorems and critical points (12 questions), increasing, decreasing and the first derivative test (10), the candidates test (6), concavity and the second derivative test (11), connecting f, f′ and f″ (8), optimization (9) and implicit relations (6). The description weights the unit and not its topics, so the counts are ours.

Graphs in words

Where the exam would print a graph of f′, a question here describes it in a sentence: where it crosses the axis, where it peaks, which way a parabola opens.

  • To justify a local extremum, name the sign change of f′. Writing f′(c) = 0 shows only a critical point.
  • On a closed interval, compare the endpoint values with the critical point values before naming an absolute extremum.
  • A point of inflection needs f″ to change sign. f″(c) = 0 alone does not make one.

Questions and answers

What is Unit 5 in AP Calculus?

Analytical Applications of Differentiation: the Mean Value and Extreme Value Theorems, increasing and decreasing intervals, the first and second derivative tests, concavity, curve sketching, optimization and implicit relations.

How much of the AP Calculus exam is Unit 5?

From 15 to 20% of the multiple-choice section of the AB exam and from 10 to 15% of the BC exam, by the College Board's course pages. On the AB exam only Unit 6 is weighted as heavily.

Is Unit 5 the same in AP Calculus AB and BC?

Yes. Every topic belongs to both courses. The course description suggests 15 to 16 class periods for the unit in AB and 10 to 11 in BC.

Is this the AP Calculus Unit 5 progress check?

No. The College Board's Progress Check for Unit 5 has about 35 multiple-choice and 3 free-response questions by the course description, and teachers assign it in AP Classroom. These questions are our own.

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