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 +1.1

3.04b Equilibrium: zero resultant moment and force6.02e Calculate KE and PE: using formulae6.05a Angular velocity: definitions
Sort by: Default | Easiest first | Hardest first
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.
Edexcel FP2 Q44
14 marks Standard +0.3
  1. Find the general solution of the differential equation $$2\frac{d^2 x}{dt^2} + 5\frac{dx}{dt} + 2x = 2t + 9.$$ [6]
  2. Find the particular solution of this differential equation for which \(x = 3\) and \(\frac{dx}{dt} = -1\) when \(t = 0\). [4] The particular solution in part (b) is used to model the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds (\(t \geq 0\)), \(P\) is \(x\) metres from the origin \(O\).
  3. Show that the minimum distance between \(O\) and \(P\) is \(\frac{1}{2}(5 + \ln 2)\) m and justify that the distance is a minimum. [4]