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CMA Foundation · Fundamentals of Business Mathematics and Statistics · Calculus - Application in Business

For the function f(x) = x^3 - 9x^2 + 24x + 5, which statement about its stationary points is correct?

The function has a local maximum at x = 2 and a local minimum at x = 4. The first derivative factors as 3(x-2)(x-4), and the second derivative 6x - 18 is negative at 2 and positive at 4.

  1. ALocal maximum at x = 2 and local minimum at x = 4Correct
  2. BLocal minimum at x = 2 and local maximum at x = 4
  3. CLocal maximum at both x = 2 and x = 4
  4. DLocal minimum at both x = 2 and x = 4

Explanation

f'(x) = 3x^2 - 18x + 24 = 3(x-2)(x-4), so x = 2 and 4. f''(x) = 6x - 18: at x = 2 it is -6 (maximum); at x = 4 it is +6 (minimum). The reversed option swaps the sign test.

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