A wire bent in the shape of a hyperbolic cosine function y = a cosh(x/x₀) is supported in a vertical plane, where \(x\) and \(y\) are the horizontal and vertical coordinates, respectively. and \(a\) and \(x_{0}\) are positive constants. A bead of mass \(m\) is threaded onto the wire and is free to slide without friction along it, and is subject to uniform gravity \(g\) directed downward.

(a) Find Lagrange's equations for the bead using \(x\) as the generalized coordinate, and

(b) find the frequency of small oscillations of the bead about the lowest point of the wire.