A right square pyramidal frame is formed from a rod of length \(4a\) by dividing the rod into eight pieces, four being of length \(x\) (to make the edges of the square base) and the other four of length \(a - x\) (to make the sloping edges). The rods are jointed together without wastage. If the volume of the pyramid is \(V\), and \(x\) may vary, show that \(\dfrac{dV}{dx} = 0\) when \(x = 0\) and when \[ 3x^2 - 10ax + 4a^2 = 0. \]

Show that, when \(x\) is equal to the smaller of the two roots of this equation,

  1. a real pyramid can be formed,
  2. \(V\) has a maximum value.