First and Second Derivative Tests Quiz

Using the first derivative test, determine the intervals on which the function f(x) = 3x^2 - 12x + 5 is increasing or decreasing.

• The function is always increasing

• The function is increasing on the interval (2, ∞) and decreasing on the interval (-∞, 2)

• The function is always decreasing

• The function f(x) = 3x^2 - 12x + 5 is increasing on the interval (-∞, 2) and decreasing on the interval (2, ∞).

Using the first derivative test, find the critical points of the function g(x) = x^3 - 6x^2 + 9x + 4 and determine the intervals of increase and decrease.

• The critical points are x = 0 and x = 2. The function increases on the interval (-∞, 0) and (2, ∞), and decreases on the interval (0, 2).

• The critical points are x = 2 and x = 4. The function increases on the interval (-∞, 2) and (4, ∞), and decreases on the interval (2, 4).

• The critical points are x = -1 and x = 5. The function increases on the interval (-∞, -1) and (5, ∞), and decreases on the interval (-1, 5).

• The critical points are x = 1 and x = 3. The function increases on the interval (-∞, 1) and (3, ∞), and decreases on the interval (1, 3).

Using the first derivative test, analyze the function h(x) = 4x^3 - 12x^2 + 6x + 2 to find the local maximum and minimum points.

• The local maximum point is (1, 4) and the local minimum point is (2, -8)

• The local maximum point is (1, 2) and the local minimum point is (2, -10)

• The local maximum point is (0, 2) and the local minimum point is (3, -10)

• The local maximum point is (2, 2) and the local minimum point is (1, -10)

Using the first derivative test, identify the intervals of increase and decrease for the function k(x) = 2x^4 - 8x^3 + 6x^2 + 3x - 1.

• The intervals of increase are (-∞, 0) and (0, 1), and the interval of decrease is (-∞, 1)

• The intervals of increase are (-∞, 0) and (1, ∞), and the interval of decrease is (0, 1).

• The intervals of increase are (-∞, 0) and (0, 1), and the interval of decrease is (1, ∞)

• The intervals of increase are (0, 1) and (1, ∞), and the interval of decrease is (-∞, 0)

Using the first derivative test, determine the critical points and intervals of increase and decrease for the function m(x) = x^4 - 4x^3 + 6x^2 - 4x + 1.

• m(x) has critical points at x = 0 and x = 3, and it increases on (-∞, 0) and (3, ∞), and decreases on (0, 3)

• m(x) has critical points at x = 1 and x = 2, and it increases on (-∞, 2) and (1, ∞), and decreases on (2, 1)

• m(x) has critical points at x = -1 and x = -2, and it increases on (-∞, -1) and (-2, ∞), and decreases on (-1, -2)

• m(x) has critical points at x = 1 and x = 2, and it increases on (-∞, 1) and (2, ∞), and decreases on (1, 2).

Using the second derivative test, determine the nature of the critical point for the function f(x) = 2x^3 - 6x^2 + 4x + 3.

• The critical point is a local minimum.

• The critical point is undefined.

• The critical point is a local maximum.

• The critical point is a point of inflection.

Using the second derivative test, analyze the function g(x) = 3x^4 - 8x^3 + 6x^2 - 12x + 5 to find the inflection points.

• The inflection points are x = 4 and x = 5

• The inflection points are x = -1 and x = -2

• The inflection points are x = 0 and x = 3

• The inflection points are x = 1 and x = 2

Using the second derivative test, find the nature of the critical point for the function h(x) = x^4 - 4x^3 + 6x^2 - 4x + 2.

• The critical point is (2, 4) and the second derivative is positive, indicating a local minimum.

• The critical point is (1, 3) and the second derivative is negative, indicating a local maximum.

• The critical point is (1, 3) and the second derivative is positive, indicating a local minimum.

• The critical point is (0, 2) and the second derivative is negative, indicating a local maximum.

Using the second derivative test, determine the concavity of the function k(x) = 4x^3 - 12x^2 + 12x - 3.

• The function k(x) = 4x^3 - 12x^2 + 12x - 3 is concave up for x < 1 and concave down for x > 1.

• The function k(x) is concave down for x < 1 and concave up for x > 1

• The function k(x) is always concave down

• The function k(x) is always concave up

Using the second derivative test, analyze the function m(x) = 3x^4 - 16x^3 + 24x^2 - 12x + 2 to find the points of inflection.

• The points of inflection are at x = -1 and x = -2

• The points of inflection are at x = 1 and x = 2

• The points of inflection are at x = 0 and x = 3

• The points of inflection are at x = 4 and x = 5