Problem from the sky
Ignoring homotheties, let the following cycloid:
x = t - sin t
y = 1 - cos t,
t in [0, 2pi].
From this, we have y' = dy/dx = (dy/dt) / (dx/dt) = (sin t) / (1 - cos t) = ... = cot (t/2), and then y'' = d(y')/dx = ... = -(1/4) [ cosec (t/2) ]^4, ..., "AND SO ON" :rofl...
Mr. Benorin, you see, this is a local problem: the final result is evaluated at x = a. Besides that, f satisfies some particular conditions, which must be considered:
(a) f(x) \neq 0, over some open interval A;
(b) f is a series of even powers;
(c) f^{(2n+1)}(a) = 0 and f^{(2n)}(a) \neq 0, n...