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At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (202

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Question: At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (2020)

Options:

  1. (1, 2)
  2. (2, 1)
  3. (0, 4)
  4. (3, 0)

Correct Answer: (1, 2)

Exam Year: 2020

Solution: 

To find local minima, we find f\'(x) = 3x^2 - 6x. Setting f\'(x) = 0 gives x = 0 and x = 2. Checking the second derivative, f\'\'(2) = 6 > 0, so (2, 1) is a local minimum.

At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (202

Practice Questions

Q1
At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (2020)
  1. (1, 2)
  2. (2, 1)
  3. (0, 4)
  4. (3, 0)

Questions & Step-by-Step Solutions

At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (2020)
  • Step 1: Write down the function f(x) = x^3 - 3x^2 + 4.
  • Step 2: Find the first derivative f'(x) to determine where the slope is zero. The first derivative is f'(x) = 3x^2 - 6x.
  • Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 6x = 0.
  • Step 4: Factor the equation: 3x(x - 2) = 0. This gives us two solutions: x = 0 and x = 2.
  • Step 5: To determine if these points are local minima or maxima, we need to find the second derivative f''(x). The second derivative is f''(x) = 6x - 6.
  • Step 6: Evaluate the second derivative at the critical points. First, check x = 0: f''(0) = 6(0) - 6 = -6 (which indicates a local maximum).
  • Step 7: Now check x = 2: f''(2) = 6(2) - 6 = 6 (which indicates a local minimum).
  • Step 8: The local minimum occurs at x = 2. To find the corresponding y-value, substitute x = 2 back into the original function: f(2) = 2^3 - 3(2^2) + 4 = 8 - 12 + 4 = 0.
  • Step 9: Therefore, the local minimum point is (2, 0).
  • Finding Local Minima – The process of determining points where a function reaches a local minimum by using first and second derivatives.
  • Critical Points – Points where the first derivative is zero or undefined, indicating potential local extrema.
  • Second Derivative Test – A method to classify critical points as local minima, local maxima, or points of inflection based on the sign of the second derivative.
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