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Determine the critical points of f(x) = x^3 - 6x^2 + 9x.

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Question: Determine the critical points of f(x) = x^3 - 6x^2 + 9x.

Options:

  1. x = 0, 3
  2. x = 1, 2
  3. x = 2, 3
  4. x = 1, 3

Correct Answer: x = 0, 3

Solution: 

Setting f\'(x) = 0 gives critical points at x = 0 and x = 3.

Determine the critical points of f(x) = x^3 - 6x^2 + 9x.

Practice Questions

Q1
Determine the critical points of f(x) = x^3 - 6x^2 + 9x.
  1. x = 0, 3
  2. x = 1, 2
  3. x = 2, 3
  4. x = 1, 3

Questions & Step-by-Step Solutions

Determine the critical points of f(x) = x^3 - 6x^2 + 9x.
  • Step 1: Write down the function f(x) = x^3 - 6x^2 + 9x.
  • Step 2: Find the derivative of the function, f'(x).
  • Step 3: Use the power rule to differentiate: f'(x) = 3x^2 - 12x + 9.
  • Step 4: Set the derivative equal to zero: 3x^2 - 12x + 9 = 0.
  • Step 5: Simplify the equation by dividing everything by 3: x^2 - 4x + 3 = 0.
  • Step 6: Factor the quadratic equation: (x - 1)(x - 3) = 0.
  • Step 7: Solve for x by setting each factor to zero: x - 1 = 0 or x - 3 = 0.
  • Step 8: Find the critical points: x = 1 and x = 3.
  • Finding Critical Points – This involves taking the derivative of the function and setting it to zero to find points where the function's slope is zero.
  • Derivative Calculation – Understanding how to correctly compute the derivative of a polynomial function.
  • Identifying Local Extrema – Recognizing that critical points can indicate local maxima, minima, or points of inflection.
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