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4 marksRoots of Equations

CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2 · Question 4(a)(iii)

The function f is given by f(x) = 6 - 4x - x^3.

Show that alpha is the only real root of the equation f(x) = 0.

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Other parts of this question

  1. 4(a)(i)Show that f is everywhere strictly decreasing.[4 marks]
  2. 4(a)(ii)Show that the equation f(x) = 0 has a real root, alpha, in the closed interval [1, 2].[4 marks]
  3. 4(b)If x_n is the nth approximation to alpha, use the Newton-Raphson method to show that the (n + 1)th approximation x_{n+1} is given by x_{n+1} = (2x_n^3 + 6) /…[8 marks]

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