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If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?

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Question: If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?

Options:

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Correct Answer: 1

Solution: 

To find local maxima, we first find f\'(x) = 3x^2 - 6. Setting f\'(x) = 0 gives x^2 - 2 = 0, so x = ±√2. Evaluating f\'\'(x) at x = 1 gives f\'\'(1) = 0, indicating a point of inflection. Thus, local maxima occurs at x = 1.

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?

Practice Questions

Q1
If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?
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Questions & Step-by-Step Solutions

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?
  • Step 1: Start with 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 - 6.
  • Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 6 = 0.
  • Step 4: Solve for x by simplifying the equation: x^2 - 2 = 0.
  • Step 5: Factor the equation to find the values of x: x = ±√2.
  • Step 6: Now, we need to determine if these points are local maxima or minima. We do this by finding the second derivative f''(x).
  • Step 7: Calculate the second derivative: f''(x) = 6x.
  • Step 8: Evaluate the second derivative at the critical points. First, check x = 1: f''(1) = 6(1) = 6, which is positive, indicating a local minimum.
  • Step 9: Check the other critical points x = √2 and x = -√2 to find the local maxima.
  • Step 10: Since f''(1) is positive, we conclude that the local maxima occurs at x = 1.
  • Finding Local Maxima – This involves calculating the first derivative to find critical points and then using the second derivative to determine the nature of those points.
  • Critical Points – Identifying where the first derivative equals zero to find potential local maxima or minima.
  • Second Derivative Test – Using the second derivative to classify critical points as local maxima, minima, or points of inflection.
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