Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These 20 problems cover AP Calculus AB Unit 5: critical numbers, increasing and decreasing intervals, concavity, extrema tests, the Mean Value and Extreme Value Theorems, curve sketching and optimization. Early problems apply a single test; later ones need several.
The last six problems are applied or multi-step, including a fence, a cardboard box and a drone, where a model must be built before it is differentiated.
Standards: AP Calculus AB Unit 5 · Level: Grade 12 AP Calculus AB
Problem 1 Absolute extrema on a closed interval ★★★
Find the absolute maximum and minimum of f(x) = x3 − 3x2 − 9x + 5 on [−2, 4].
Problem 2 Increasing and decreasing intervals ★★★
Find where g(x) = x4 − 8x2 is increasing and where it is decreasing.
Problem 3 First Derivative Test with an exponential ★★★
Let h(x) = xe−2x. Find its local extremum and classify it.
Problem 4 Concavity and inflection points ★★★
Let f(x) = x4 − 6x2 + 4x. Find the intervals of concavity and the inflection points (give x-values).
Problem 5 Second Derivative Test ★★★
Classify the critical points of f(x) = x3 − 12x + 7 using the sign of the second derivative.
Problem 6 Mean Value Theorem ★★★
Verify the hypotheses of the Mean Value Theorem for f(x) = x2 − 6x + 1 on [1, 7] and find every guaranteed value c.
Problem 7 When the Mean Value Theorem does not apply ★★★
Let f(x) = |x − 2| on [0, 5]. Explain which hypothesis of the Mean Value Theorem fails, and show that no number c has the matching slope.
Problem 8 Locating an inflection point ★★★
Find the inflection point of f(x) = 2x3 − 15x2 + 36x.
Problem 9 Reading the graph of a derivative ★★★
The graph of f′(x) is shown. On which intervals is f increasing? Where does f have local maxima and minima?
Problem 10 Inflection points from a derivative formula ★★★
If f′(x) = (x + 2)(x − 1)(x − 4)4, find the x-coordinates of the inflection points of f.
Problem 11 Absolute extrema with a reciprocal ★★★
Find the absolute extrema of f(x) = x + 4x on [1, 5].
Problem 12 Fencing a pen against a barn ★★★
A farmer has 180 feet of fence to enclose a rectangular pen. One side is a barn and needs no fence. Find the dimensions that maximize the area and state the maximum area.
Problem 13 Open box from cardboard ★★★
Squares of side x are cut from the corners of a 30-inch by 20-inch sheet and the sides are folded up to form an open box. Find x that maximizes the volume, and the maximum volume to the nearest cubic inch.
Problem 14 Max and min on an ellipse ★★★
On the ellipse x2 + 4y2 = 16, find the points where the tangent line is horizontal and use the second derivative to classify them.
Problem 15 Connecting f, f′ and f″ ★★★
A continuous function f has f′(x) > 0 on (−∞, −1), f′(x) < 0 on (−1, 3), f′(x) > 0 on (3, ∞), and f″ < 0 on (−∞, 1), f″ > 0 on (1, ∞). Locate all local extrema and inflection points and describe the sketch.
Problem 16 Ticket price and revenue ★★★
A theater finds that at a ticket price of p dollars it sells 500 − 20p tickets. The price must be between $5 and $20. Use the Candidates Test to find the price that maximizes revenue.
Problem 17 Minimizing the surface of a can ★★★
A cylindrical can must hold 500 cm3. Determine r and h giving the least surface area, to the nearest 0.1 cm.
Problem 18 Mean Value Theorem for a drone ★★★
A drone’s altitude h(t), in meters, is differentiable on [0, 8] with h(0) = 15 and h(8) = 95. Must the drone have a vertical speed of exactly 10 m/s at some moment? Must it have exactly 12 m/s?
Problem 19 Complete analysis of a quartic ★★★
Analyze f(x) = x4 − 4x3: find critical numbers, intervals of increase, local extrema, concavity and inflection points.
Problem 20 Finding coefficients from conditions ★★★
Let f(x) = x3 + ax2 + bx + c. The graph has an inflection point at x = 2 and a critical number at x = −1, and f(0) = −7. Find a, b, c and classify both critical points.
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