f(x) is a differentiable function and g(x) is a doubly differentiable function such that |f(x)| <= 2 and f'(x) = g(x). If [f(0)]^2 + [g(0)]^2 = 9, prove that there exists some c in (-3, 3) such that g(x), g''(c) < 0.

A ray of light traveling in air is incident at grazing angle (incident angle = 90 degrees) n a long rectangular slab of a transparent medium of thickness t = 1.0 m. The point of incidence is the origin A(0,0). The medium has a variable index of refraction ny) given by n(y) = [ky^(3/2)+1]^(1/2) where k = 1.0 (meter)^(-3/2). The refractive index of air is 1.0. Obtain an equation for the trajectory y(x) of the ray in the medium.

Is there a strictly increasing function f : R -> R such that f'(x) = f(f(x)) for all x?