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7 marksThe Real Number System – ℝ

CAPE Pure Mathematics Unit 1 · May/June 2003 · Paper 2 · Question 5(a)

Given that f(x) = x³ - 5x² + 3x.

Show that f(x) = 0 possesses a root in the interval [1/2, 1]. By considering suitable values of x greater than 1, show that there is another root of f(x) = 0 greater than 1.

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Other parts of this question

  1. 5(b)(i)Find the coordinates of the stationary points of f(x).[6 marks]
  2. 5(b)(ii)Find the second derivative of f(x), and hence, determine which stationary point is a local maximum and which is a local minimum.[5 marks]
  3. 5(c)If y = 1/(x² + 2), show that d²y/dx² = 2(3x² - 2)y³.[7 marks]

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