A particle constrained to move along the x-axis has a potential energy that varies qualitatively with position as follows: the curve is symmetrical about x=0, attains a minimum value at x=0, and rises smoothly on either side, appearing parabolic for small ∣x∣ and steeper for larger ∣x∣.
a. (i) Identify the nature of equilibrium at x=0 and justify your answer with reference to the given shape of the potential energy curve.
b. (ii) Explain how the sign of the slope of the curve on either side of x=0 determines the direction of the force acting on the particle.
c. (iii) Suppose elsewhere on the x-axis the same potential energy curve possessed an isolated local maximum. State the type of equilibrium that would exist at such a point and give one reason for your choice.