AP Calculus AB: Increasing/Decreasing and First Derivative Test (Drill 1)

Question 1. Let \( f \) be a differentiable function with derivative \( f'(x) = (x-1)(x+3) \). On which of the following intervals is \( f \) increasing?

• A) \( (-3, 1) \)
• B) \( (-\infty, -3) \) and \( (1, \infty) \) ✓
• C) \( (-\infty, 1) \)
• D) \( (-3, \infty) \)

Explanation: Choice B is correct. \( f \) is increasing where \( f'(x) > 0 \). The critical points are \( x = -3 \) and \( x = 1 \). On \( (-\infty, -3) \), both factors are negative, so \( f'(x) > 0 \). On \( (-3, 1) \), \( (x+3) > 0 \) but \( (x-1) < 0 \), so \( f'(x) 0 \). Choice A is incorrect because \( (-3, 1) \) is where \( f \) is decreasing. Choice C is incorrect because it includes \( (-3, 1) \), where \( f'(x) < 0 \). Choice D is incorrect because it also includes \( (-3, 1) \), where \( f \) is decreasing.

Question 2. The function \( g \) is continuous on \( (-\infty, \infty) \). The sign of \( g'(x) \) is positive for \( x 2 \). Which of the following must be true?

• A) \( g \) has a relative minimum at \( x = 2 \).
• B) \( g \) has a relative maximum at \( x = 2 \). ✓
• C) \( g \) has an absolute maximum at \( x = 2 \).
• D) \( g \) has no extremum at \( x = 2 \) because \( g'(2) \) may not exist.

Explanation: Choice B is correct. Since \( g'(x) \) changes from positive on the left of \( x = 2 \) to negative on the right of \( x = 2 \), the First Derivative Test guarantees that \( g \) has a relative maximum there. Choice A is incorrect because a relative minimum requires \( g' \) to change from negative to positive, the opposite of what is given. Choice C is incorrect because the First Derivative Test identifies only relative extrema; confirming an absolute maximum requires comparing all critical points and endpoints on a closed interval. Choice D is incorrect because the First Derivative Test requires only that \( g \) be continuous at \( x = 2 \) and that \( g' \) changes sign there, both conditions are satisfied regardless of whether \( g'(2) \) is zero or undefined.

Question 3. Let \( f(x) = x^3 - 6x^2 + 9x + 1 \). Which of the following correctly identifies the relative extrema of \( f \)?

• A) Relative maximum at \( x = 1 \); relative minimum at \( x = 3 \) ✓
• B) Relative minimum at \( x = 1 \); relative maximum at \( x = 3 \)
• C) Relative maximum at \( x = 3 \) only
• D) Relative minimum at \( x = 1 \) only

Explanation: Choice A is correct. \( f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3) \), giving critical points at \( x = 1 \) and \( x = 3 \). On \( (-\infty, 1) \), \( f'(x) > 0 \); on \( (1, 3) \), \( f'(x) 0 \). Since \( f' \) changes from positive to negative at \( x = 1 \), there is a relative maximum there. Since \( f' \) changes from negative to positive at \( x = 3 \), there is a relative minimum there. Choice B is incorrect because it reverses both classifications, a positive-to-negative sign change indicates a maximum, not a minimum. Choice C is incorrect because it omits the relative maximum at \( x = 1 \). Choice D is incorrect because it omits the relative minimum at \( x = 3 \).

Question 4. Let \( h(x) = x^3 \). Which of the following correctly describes the behavior of \( h \) at \( x = 0 \)?

• A) \( x = 0 \) is a relative minimum because \( h'(0) = 0 \).
• B) \( x = 0 \) is a relative maximum because \( h'(0) = 0 \).
• C) \( x = 0 \) is neither a relative maximum nor a relative minimum because \( h'(x) \) does not change sign at \( x = 0 \). ✓
• D) \( x = 0 \) is a relative minimum because \( h \) is concave up for \( x > 0 \).

Explanation: Choice C is correct. \( h'(x) = 3x^2 \), so \( h'(x) > 0 \) for \( x 0 \) for \( x > 0 \). Since \( h'(x) \) does not change sign at \( x = 0 \), the First Derivative Test confirms there is no relative extremum there. This is the classic example of a critical point that is neither a maximum nor a minimum. Choice A is incorrect because \( h'(0) = 0 \) identifies \( x = 0 \) as a critical point but does not guarantee an extremum, a sign change in \( h' \) is required. Choice B is incorrect for the same reason. Choice D is incorrect because concavity alone does not determine the presence of a relative minimum; the First Derivative Test requires a sign change in \( h' \), which does not occur here.

Question 5. The function \( f \) is differentiable on \( (-\infty, \infty) \). The table below shows the sign of \( f'(x) \) on selected intervals.

Interval	(-∞, -2)	(-2, 0)	(0, 5)	(5, ∞)
Sign of f'(x)	−	−	+	−

Which of the following correctly describes the relative extrema of \( f \)?

• A) \( f \) has a relative minimum at \( x = -2 \).
• B) \( f \) has a relative maximum at \( x = 0 \) and a relative minimum at \( x = 5 \).
• C) \( f \) has a relative minimum at \( x = 0 \) and a relative maximum at \( x = 5 \). ✓
• D) \( f \) has a relative minimum at \( x = -2 \) and a relative maximum at \( x = 0 \).

Explanation: Choice C is correct. At \( x = -2 \): \( f'(x) \) is negative on both sides, so there is no sign change and no extremum. At \( x = 0 \): \( f' \) changes from negative to positive, so \( f \) has a relative minimum. At \( x = 5 \): \( f' \) changes from positive to negative, so \( f \) has a relative maximum. Choice A is incorrect because \( f'(x) \) does not change sign at \( x = -2 \); it is negative on both adjacent intervals, so no extremum exists there. Choice B is incorrect because it reverses the classifications at \( x = 0 \) and \( x = 5 \): a negative-to-positive transition yields a minimum, not a maximum. Choice D is incorrect because it wrongly assigns an extremum at \( x = -2 \) (no sign change occurs there) and reverses the classification at \( x = 0 \).