An application of calculus

Nancy Neutron is living in a uranium nucleus that is undergoing fission. Nancy's potential energy as a function of position can be approximated by PE = x⁴ - x², where all the units and numerical constants have been suppressed for simplicity. Use calculus to locate the equilibrium points, and determine whether they are stable or unstable.

The equilibrium points occur where the PE is at a minimum or maximum, and minima and maxima occur where the derivative (which equals minus the force on Nancy) is zero. This derivative is dPE/dx = 4x³ - 2x, and setting it equal to zero, we have x = 0,±1/√2. Minima occur where the second derivative is positive, and maxima where it is negative. The second derivative is 12x² - 2, which is negative at x = 0 (unstable) and positive at x = ±1/√2 (stable). Interpretation: the graph of the PE is shaped like a rounded letter ‘W,' with the two troughs representing the two halves of the splitting nucleus. Nancy is going to have to decide which half she wants to go with.