Question: A particle moving along the x-axis has potential energy at the point x given by V (x). The potential energy has a minimum at x = 0.
(a) Write the Taylor polynomial of degree 2 for V about x = 0. What can you say about the signs of the coefficients of each of the terms of the Taylor polynomial?
(b) The force on the particle at the point x is given by -V' (x). For small x, show that the force on the particle is approximately proportional to its distance from the origin. What is the sign of the proportionality constant? Describe the direction in which the force points.