AP Calculus AB: Curve Sketching and Connecting f, f’, f” (Drill 1)

Question 1. The graph of \( f' \) is positive on \( (-2, 1) \), equals zero at \( x = 1 \), and is negative on \( (1, 4) \). Which of the following correctly describes \( f \)?

• A) \( f \) has a relative minimum at \( x = 1 \)
• B) \( f \) has a relative maximum at \( x = 1 \) ✓
• C) \( f \) is concave up on \( (-2, 4) \)
• D) \( f \) is decreasing on \( (-2, 1) \)

Explanation: Choice B is correct. Since \( f'(x) > 0 \) on \( (-2, 1) \), \( f \) is increasing there; since \( f'(x) 0 \) on \( (-2, 1) \) means \( f \) is increasing, not decreasing, on that interval.

Question 2. The function \( f \) is continuous and twice differentiable. \( f''(x) > 0 \) on \( (0, 3) \) and \( f''(x) < 0 \) on \( (3, 6) \). Which of the following must be true?

• A) \( f \) has a relative maximum at \( x = 3 \)
• B) \( f \) has a relative minimum at \( x = 3 \) under the stated conditions
• C) \( f \) has an inflection point at \( x = 3 \) ✓
• D) \( f'(3) = 0 \)

Explanation: Choice C is correct. Since \( f \) is continuous and twice differentiable, concavity is well-defined on both sides of \( x = 3 \). Because \( f''(x) > 0 \) on \( (0, 3) \), \( f \) is concave up there, and because \( f''(x) < 0 \) on \( (3, 6) \), \( f \) is concave down there. This sign change in \( f'' \) confirms that \( f \) has an inflection point at \( x = 3 \). Choice A is incorrect because relative extrema of \( f \) are determined by sign changes in \( f' \), not \( f'' \); a sign change in \( f'' \) indicates an inflection point, not a relative maximum. Choice B is incorrect for the same reason, a sign change in \( f'' \) does not imply \( f' \) changes sign. Choice D is incorrect because \( f' \) need not equal zero at an inflection point; \( f \) can change concavity while still increasing or decreasing.

Question 3. A function \( f \) is continuous on \( [-3, 3] \) with the following properties: \( f'(x) 0 \) on \( (0, 3) \), and \( f''(x) < 0 \) on \( (-3, 3) \). Which of the following correctly describes the graph of \( f \) on \( [-3, 3] \)?

• A) Decreasing then increasing, concave up throughout
• B) Decreasing then increasing, concave down throughout ✓
• C) Increasing then decreasing, concave up throughout
• D) Increasing then decreasing, concave down throughout

Explanation: Choice B is correct. Since \( f' 0 \) on \( (0, 3) \), \( f \) decreases then increases, with a relative minimum at \( x = 0 \). Since \( f'' < 0 \) throughout the interval, the graph is concave down everywhere. Choice A is incorrect because \( f'' < 0 \) means concave down, not concave up. Choice C is incorrect because the sign pattern of \( f' \) (negative then positive) indicates decreasing then increasing, not increasing then decreasing. Choice D is incorrect because increasing then decreasing would require \( f' \) to go from positive to negative, the opposite of what is given.

Question 4. The graph of \( f \) is concave up on \( (-1, 2) \) and concave down on \( (2, 5) \). Which of the following must be true?

• A) \( f'(2) = 0 \)
• B) \( f \) has a relative extremum at \( x = 2 \)
• C) \( f' \) is increasing on \( (-1, 2) \) and decreasing on \( (2, 5) \) ✓
• D) \( f' \) is negative on \( (-1, 2) \) and positive on \( (2, 5) \) for the given graph

Explanation: Choice C is correct. When \( f \) is concave up, \( f'' > 0 \), which means \( f' \) is increasing. When \( f \) is concave down, \( f'' < 0 \), which means \( f' \) is decreasing. So \( f' \) increases on \( (-1, 2) \) and decreases on \( (2, 5) \). Choice A is incorrect because a change in concavity does not require \( f'(2) = 0 \); \( f \) can change concavity while the tangent line has a nonzero slope. Choice B is incorrect because a change in concavity does not force a relative extremum, while \( x = 2 \) may be an inflection point of \( f \), a relative extremum requires \( f' \) to change sign, which is not implied here. Choice D is incorrect because the sign of \( f' \) indicates whether \( f \) is increasing or decreasing, which is entirely independent of concavity.

Question 5. The table below gives values of \( f'(x) \) at selected points for a twice-differentiable function \( f \) whose derivative \( f' \) is strictly increasing on \( [0, 4] \).

x	0	1	2	3	4
f'(x)	−3	−1	0	2	5

Which of the following conclusions is supported by this data?

• A) \( f \) has a relative minimum at \( x = 2 \) ✓
• B) \( f \) is concave down on \( (0, 4) \)
• C) \( f \) has an inflection point at \( x = 2 \)
• D) \( f \) has a relative maximum at \( x = 2 \)

Explanation: Choice A is correct. Since \( f' \) is strictly increasing on \( [0, 4] \), it crosses zero exactly once, at \( x = 2 \). To the left, \( f'(x) 0 \) (as confirmed at \( x = 3 \) and \( x = 4 \)). By the First Derivative Test, this sign change from negative to positive means \( f \) has a relative minimum at \( x = 2 \). Choice B is incorrect because \( f' \) is strictly increasing on \( (0, 4) \), which means \( f'' > 0 \), so \( f \) is concave up, not concave down. Choice C is incorrect because an inflection point requires \( f'' \) to change sign; since \( f' \) is strictly increasing throughout, \( f'' > 0 \) everywhere on the interval and there is no inflection point. Choice D is incorrect because a relative maximum requires \( f' \) to change from positive to negative, the opposite of what the data shows.