Practice a College Board-style free response question on extrema, increasing/decreasing intervals, and absolute extrema. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 5 · Analytical Applications
Let f be the function defined by f(x) = x³ − 6x² + 9x + 2 on the closed interval [0, 5].
A
Find the x-coordinates of all critical points of f, and determine the open intervals on which f is increasing and decreasing. Justify your answer.
✓ Model answer
f′(x) = 3x² − 12x + 9 = 3(x² − 4x + 3) = 3(x − 1)(x − 3). Critical points at x = 1 and x = 3. Sign of f′: positive on (0, 1), negative on (1, 3), positive on (3, 5). So f is increasing on (0, 1) and (3, 5) and decreasing on (1, 3).
Why it scores: Correct f′, correct critical points, and increasing/decreasing intervals justified by the sign of f′. A sign chart or explicit sign statements earn the justification.
B
Classify each critical point as a relative maximum or minimum. Justify using the First Derivative Test.
✓ Model answer
At x = 1, f′ changes from positive to negative, so f has a relative maximum. At x = 3, f′ changes from negative to positive, so f has a relative minimum.
Why it scores: Uses the sign change of f′ at each critical point to justify the classification. Stating the conclusion without the sign change loses the justification point.
C
Find the absolute maximum and absolute minimum values of f on [0, 5]. Justify your answer.
✓ Model answer
By the Candidates Test, evaluate f at the critical points and endpoints. f(0) = 2, f(1) = 1 − 6 + 9 + 2 = 6, f(3) = 27 − 54 + 27 + 2 = 2, f(5) = 125 − 150 + 45 + 2 = 22. The absolute maximum is 22 at x = 5 and the absolute minimum is 2, attained at x = 0 and x = 3.
Why it scores: Evaluates f at all critical points and both endpoints and identifies the largest and smallest values. Skipping an endpoint (especially x = 5) would miss the true maximum.
How to score points on AP Calculus FRQs
Justify with the sign of the derivative, not just a value — a sign chart is your friend.
Use the First Derivative Test (sign change of f′) to classify relative extrema.
Run the Candidates Test for absolute extrema — never forget the endpoints.
Concavity comes from f″; inflection points need an actual sign change.
Answer the verb. "Find" needs the computation; "justify" needs the reasoning.