Applications of Differentiation - Interpreting Gradients and Concavity

Applications of Differentiation - Interpreting Gradients and Concavity

The diagram below shows the curve \(y=\mathrm{{f}'}\left( x \right).\)

  1. State the \(x\)-coordinates of the stationary points for the curve \(y=\mathrm{f}\left( x \right).\) Hence determine the nature of these stationary points.
  2. State the maximal range of values of \(x\) such that the curve \(y=\mathrm{f}\left( x \right)\) is,
    1. concave downwards,
    2. strictly decreasing.

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Answer:(a) \(x=1\) maximum, \(x=3\) minimum, \(x=5\) maximum (b)(i) \(x<0\) or \(0<x<2\) or \(x>4\) (b)(ii) \(x<0\) or \(1<x<3\) or \(x>5\)

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