Determine the maximum and minimum of each function on the given interval.
a) ���� = 2x^3�� − 9x^2�� ,−2 ≤ x� ≤ 4
b) ���� = 12x� − x^3�� , � x∈ �[−3,5]
To determine the maximum and minimum of each function on the given interval, we need to find the critical points and endpoints of the interval.
a) For the function � = 2x^3 - 9x^2, -2 ≤ x ≤ 4:
Step 1: Find the critical points by taking the derivative of the function:
�' = 6x^2 - 18x
Step 2: Set the derivative equal to zero and solve for x to find the critical points:
6x^2 - 18x = 0
6x(x - 3) = 0
This equation has two solutions: x = 0 and x = 3. These are potential critical points.
Step 3: Evaluate the function at the critical points, as well as at the endpoints of the interval:
• x = -2: �(-2) = 2(-2)^3 - 9(-2)^2 = -40
• x = 0: �(0) = 2(0)^3 - 9(0)^2 = 0
• x = 3: �(3) = 2(3)^3 - 9(3)^2 = 9
• x = 4: �(4) = 2(4)^3 - 9(4)^2 = -160
Step 4: Compare these values to determine the maximum and minimum:
- The maximum value is 9, which occurs at x = 3.
- The minimum value is -160, which occurs at x = 4 and x = -2.
b) For the function � = 12x - x^3, x ∈ [-3, 5]:
Step 1: Find the critical points by taking the derivative of the function:
�' = 12 - 3x^2
Step 2: Set the derivative equal to zero and solve for x to find the critical points:
12 - 3x^2 = 0
3x^2 = 12
x^2 = 4
x = ±2
This equation has two solutions: x = 2 and x = -2. These are potential critical points.
Step 3: Evaluate the function at the critical points, as well as at the endpoints of the interval:
• x = -3: �(-3) = 12(-3) - (-3)^3 = -27
• x = -2: �(-2) = 12(-2) - (-2)^3 = -8
• x = 2: �(2) = 12(2) - (2)^3 = 20
• x = 5: �(5) = 12(5) - (5)^3 = -25
Step 4: Compare these values to determine the maximum and minimum:
- The maximum value is 20, which occurs at x = 2.
- The minimum value is -27, which occurs at x = -3.
In summary:
a) The maximum value of the function ���� on the interval [-2, 4] is 9, which occurs at x = 3. The minimum value is -160, which occurs at x = -2 and x = 4.
b) The maximum value of the function ���� on the interval [-3, 5] is 20, which occurs at x = 2. The minimum value is -27, which occurs at x = -3.