the first and second derivative tests and the candidates test

Review the first and second derivative tests and the candidates test for AP Calculus AB and try 2 free AP-style practice questions. 22 questions on this topic in AP Calculus AB · Unit 5 · Analytical Applications of Differentiation.

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Question 1

Suppose $f'(x)$ changes sign from negative to positive at $x = 3$ , so $f$ has a local minimum at $x = 3$ . If the entire graph of $f'$ is then reflected across the $x$ -axis, the point $x = 3$ becomes

Question 2

Let $f(x) = \frac{1}{3}x^3 - 5x^2 + 21x$ on the closed interval $[0, 8]$ . At what value of $x$ does $f$ attain its absolute maximum?

20 more questions on this topic

Full practice for AP Calculus AB · Unit 5 · Analytical Applications of Differentiation: timed tests, step-by-step explanations for every answer, and AI grading for free-response questions.

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Answer, then see why

The price of gasoline rises 10% and the quantity demanded falls 5%. What happens to total revenue?

Not quite. The answer is B.

Why the other options are wrong

AI grading for free-response

Write your FRQ answer and get it scored point by point against the AP rubric, with the line from your answer that earned or missed each point.

Question 13 / 4 pts

Part a1 / 1 pts

Point 1: Earned

94% sure

Reason: Correctly identifies demand for gasoline as inelastic.

Evidence: "The price of gasoline rises and the quantity falls only a little."

Part b2 / 3 pts

Point 1: Earned

91% sure

Reason: Graph shows relatively inelastic demand.

Evidence: Draws a steep demand curve.

Point 2: Earned

88% sure

Reason: Correct new equilibrium price.

Evidence: Shifts supply left, P1 → P2.

Point 3: Not earned

82% sure

Reason: With inelastic demand, a price increase raises total revenue.

Evidence: "Revenue goes down."

The first and second derivative tests and the candidates test: AP Calculus AB Practice Questions | S&D Academy