This question asks where function h is increasing, which occurs when h'(x) = (x+3)²(x-2) > 0. The derivative equals zero at x = -3 (with multiplicity 2) and x = 2, but only x = 2 changes the sign of h'(x) since (x+3)² is always non-negative. For x < 2, the factor (x-2) is negative while (x+3)² ≥ 0, making h'(x) ≤ 0 (strictly negative except at x = -3). For x > 2, the factor (x-2) is positive and (x+3)² > 0, making h'(x) > 0. Students might incorrectly select option B by treating x = -3 as a sign-changing point, but even-powered factors don't change sign. The crucial insight is recognizing that squared factors only create horizontal tangents without sign changes, so only odd-powered factors determine where the derivative changes from positive to negative.

This problem asks where r is decreasing, requiring r'(x) = (x+2)²(x-6) < 0. The squared factor (x+2)² is always non-negative and equals zero only at x = -2, while (x-6) changes sign at x = 6. For x < -2: (x+2)² > 0 and (x-6) < 0, so r'(x) < 0; for -2 < x < 6: (x+2)² > 0 and (x-6) < 0, so r'(x) < 0; for x > 6: (x+2)² > 0 and (x-6) > 0, so r'(x) > 0. Note that at x = -2, r'(-2) = 0·(-8) = 0, so r is neither increasing nor decreasing at this point, but r continues to decrease on both sides. Choice E incorrectly separates the decreasing interval at x = -2, but since r'(x) < 0 on both sides, the function decreases throughout (-∞,6). Remember that squared factors don't cause sign changes in derivatives, only the odd-powered factors do.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is increasing, identify where f'(x) > 0, as a positive derivative indicates the function's slope is upward. The critical points are x = -2 and x = 1, dividing the real line into intervals (-∞, -2), (-2, 1), and (1, ∞). Testing signs shows f'(x) > 0 in (-∞, -2) and (1, ∞), while f'(x) < 0 in (-2, 1). A tempting distractor like (-∞, 1) fails because it includes (-2, 1) where the derivative is negative, incorrectly suggesting increase there. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is decreasing, identify where f'(x) < 0, as a negative derivative indicates the function's slope is downward. The critical points are x = 0 and x = 4, dividing the real line into intervals (-∞, 0), (0, 4), and (4, ∞). Testing signs shows f'(x) < 0 in (-∞, 0) and (4, ∞), while f'(x) > 0 in (0, 4). A tempting distractor like (-∞, 4) fails because it includes (0, 4) where the derivative is positive, incorrectly suggesting decrease there. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is increasing, identify where f'(x) > 0, as a positive derivative indicates the function's slope is upward. The critical points are x = -1 (multiplicity 2) and x = 4, dividing the real line into intervals (-∞, -1), (-1, 4), and (4, ∞). Testing signs shows f'(x) > 0 in (4, ∞), while f'(x) < 0 in (-∞, -1) and (-1, 4). A tempting distractor like (-∞, 4) fails because the derivative is negative there, incorrectly suggesting increase instead of decrease. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is increasing, identify where f'(x) > 0, as a positive derivative indicates the function's slope is upward. The critical point is x = 2 (zero in numerator), with domain x > 0 and √x > 0, so sign follows (2 - x). Testing shows f'(x) > 0 in (0, 2), while f'(x) < 0 in (2, ∞). A tempting distractor like (2, ∞) fails because the derivative is negative there, incorrectly suggesting increase instead of decrease. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is decreasing, identify where f'(x) < 0, as a negative derivative indicates the function's slope is downward. The critical points are x = -2 and x = 1 (zeros, with $e^x$ always positive), dividing the real line into intervals (-∞, -2), (-2, 1), and (1, ∞). Testing signs shows f'(x) < 0 in (-2, 1), while f'(x) > 0 in (-∞, -2) and (1, ∞). A tempting distractor like (-∞, -2)∪(1, ∞) fails because the derivative is positive there, incorrectly suggesting decrease instead of increase. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is increasing, identify where f'(x) > 0, as a positive derivative indicates the function's slope is upward. The critical points are x = -3, x = 2 (zeros), and x = 1 (undefined), dividing the real line into intervals (-∞, -3), (-3, 1), (1, 2), and (2, ∞). Testing signs shows f'(x) > 0 in (-3, 1) and (2, ∞), while f'(x) < 0 in (-∞, -3) and (1, 2). A tempting distractor like (-∞, -3)∪(1, 2) fails because the derivative is negative there, incorrectly suggesting increase instead of decrease. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is decreasing, identify where f'(x) < 0, as a negative derivative indicates the function's slope is downward. The critical point is x = 3 (undefined), with sign determined by -(x+2) over positive $(x-3)^2$, negative when x > -2. Testing intervals shows f'(x) < 0 in (-2, 3) and (3, ∞), while f'(x) > 0 in (-∞, -2). A tempting distractor like (-∞, -2)∪(3, ∞) fails because it includes (-∞, -2) where the derivative is positive, incorrectly suggesting decrease there. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.

This question tests the skill of determining intervals on which a function is increasing or decreasing using its first derivative. To find where f is decreasing, identify where f'(x) < 0, as a negative derivative indicates the function's slope is downward. The critical points are x = -4, x = 2 (zeros), and x = 0 (undefined), dividing the real line into intervals (-∞, -4), (-4, 0), (0, 2), and (2, ∞). Testing signs shows f'(x) < 0 in (-∞, -4) and (0, 2), while f'(x) > 0 in (-4, 0) and (2, ∞). A tempting distractor like (-4, 2) fails because it includes (-4, 0) where the derivative is positive, incorrectly suggesting decrease there. Always create a sign chart for the derivative by identifying roots and testing intervals to determine where the function is increasing or decreasing.