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MTH 102 MTH 102 Past Questions Objective Question
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Question

Evaluate the minimum value of y = x³ - 3x² + 1.
Options
A <code>2</code>
B <code>-3</code>
Correct Answer
C <code>3</code>
D <code>-2</code>
Correct Answer

Option B is the correct answer.

Detailed Explanation

To find the minimum value of the function, we follow these steps:
  1. Find the first derivative y':y' = d/dx(x³ - 3x² + 1) = 3x² - 6x
  2. Set the first derivative to zero to find critical points:3x² - 6x = 03x(x - 2) = 0The critical points are x = 0 and x = 2.
  3. Find the second derivative y'':y'' = d/dx(3x² - 6x) = 6x - 6
  4. Use the second derivative test to classify the critical points:
    • At x = 0: y''(0) = 6(0) - 6 = -6. Since -6 < 0, x = 0 corresponds to a local maximum.
    • At x = 2: y''(2) = 6(2) - 6 = 12 - 6 = 6. Since 6 > 0, x = 2 corresponds to a local minimum.
  5. Calculate the function's value at the local minimum x = 2:y(2) = (2)³ - 3(2)² + 1y(2) = 8 - 3(4) + 1y(2) = 8 - 12 + 1 = -4 + 1 = -3
The minimum value of the function is -3.

Hint

Find the critical points by setting the first derivative to zero. Then use the second derivative test to determine whether each critical point is a local maximum or minimum. Substitute the x-value of the local minimum back into the original function.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 438 of 930
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