Show that with this form for the system path, I is an extremum for nonvanishing x only if aj = 0, and only if?

Question

The one-dimensional harmonic oscillator has the lagrangian L = m(dx/dt)^2 /2 - kx^2 /2. Suppose you didn't know the solution to the motion, but realized that the motion must be periodic and therefore could be described by a fourier series of the form x(t) =sumation of aj cosjwt , ( taking t = 0 at a turning point) where w is the unknown angular frequency of the motion. Consider the action integral I for two points, t1 and t2 seprated by the period T=2pi /w .

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Your question is a bit unclear. Could you elaborate?