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The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?

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Question: The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?

Options:

  1. 0
  2. 1
  3. 2
  4. 3

Correct Answer: 1

Solution: 

Finding f\'(x) = 3x^2 - 12x + 9. Setting f\'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.

The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?

Practice Questions

Q1
The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
  1. 0
  2. 1
  3. 2
  4. 3

Questions & Step-by-Step Solutions

The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
  • Step 1: Write down the function f(x) = x^3 - 6x^2 + 9x.
  • Step 2: Find the first derivative f'(x) by differentiating f(x). This gives f'(x) = 3x^2 - 12x + 9.
  • Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 12x + 9 = 0.
  • Step 4: Simplify the equation by dividing everything by 3: x^2 - 4x + 3 = 0.
  • Step 5: Factor the quadratic equation: (x - 1)(x - 3) = 0.
  • Step 6: Solve for x to find critical points: x = 1 and x = 3.
  • Step 7: To determine if these points are local extrema, find the second derivative f''(x) = 6x - 12.
  • Step 8: Evaluate the second derivative at the critical points: f''(1) = 6(1) - 12 = -6 (local maximum) and f''(3) = 6(3) - 12 = 6 (local minimum).
  • Step 9: Conclude that there is one local maximum at x = 1 and one local minimum at x = 3.
  • Local Extrema – Understanding how to find local maxima and minima using the first and second derivative tests.
  • Derivative Calculation – Calculating the first derivative to find critical points where local extrema may occur.
  • Second Derivative Test – Using the second derivative to determine the nature of the critical points found.
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