Question

Find the absolute maximum and absolute minimum values of 2X³ - 3X² on the interval [-1. 3]

Answer 1

The absolute maximum value of the function 2X³ - 3X² on the interval [-1, 3] is 27 at X = 3, and the absolute minimum value is -5 at X = -1.

Step-by-step explanation:

To find the absolute maximum and minimum values of the function 2X³ - 3X² on the interval [-1, 3], you can follow these steps:

1. Differentiate the function to find the critical points: f'(x) = 6X² - 6X.
2. Set the derivative equal to zero and solve for X to find the critical points: 0 = 6X(X - 1), which gives X = 0 and X = 1.
3. Evaluate the function at the critical points and the endpoints of the interval: f(-1), f(0), f(1), and f(3).
4. Compare the values to determine the absolute maximum and minimum.

Evaluating the function at the critical points and endpoints gives:

• f(-1) = 2(-1)³ - 3(-1)² = -2 - 3 = -5
• f(0) = 2(0)³ - 3(0)² = 0
• f(1) = 2(1)³ - 3(1)² = 2 - 3 = -1
• f(3) = 2(3)³ - 3(3)² = 54 - 27 = 27

Therefore, the absolute maximum value is 27 at X = 3, and the absolute minimum value is -5 at X = -1.