Find turning points or extrema

A question is this type if and only if it asks to find and justify the location of maximum, minimum, or turning points of the particular solution.

2 questions · Challenging +1.5

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Edexcel M4 2005 January Q5
10 marks Challenging +1.2
  1. Show that, when \(\angle B F O = \theta\), the potential energy of the system is $$\frac { 1 } { 10 } m g a ( 8 \cos \theta - 5 ) ^ { 2 } - 2 m g a \cos ^ { 2 } \theta + \text { constant } .$$
  2. Hence find the values of \(\theta\) for which the system is in equilibrium.
  3. Determine the nature of the equilibrium at each of these positions.
OCR M4 2015 June Q5
15 marks Challenging +1.8
  1. Taking \(H\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g \left( 2 \lambda r \cos \theta - 2 r \cos ^ { 2 } \theta - \lambda a \right)$$
  2. Find the set of possible values of \(\lambda\) so that there is more than one position of equilibrium.
  3. For the case \(\lambda = \frac { 3 } { 2 }\), determine whether each equilibrium position is stable or unstable.