`U = 2x^(3) - 2x^(3) - 6x^(2) + 5x`, where x is in metres and U is the potential energy which depends on x. Selsct the correct option.

Options

A. There are two stable equilibrium positions, at `x = (1 +- (1)/(sqrt6)) m`
B. `x = (1 + (1)/(sqrt6)) m` is stable equilibrium position.
C. `x = (1 - (1)/(sqrt6)) m` is stable equilibrium position.
D. `x = (1 + (1)/(sqrt6)) m` is unstable equilibrium position.

Answer

B

Explanation

`U = 2x^(3) - 6x^(2) + 5x`
at eq., `(dU)/(dx) = 6 x^(2) - 12 x + 5 = 0`
`x = 1 pm (1)/(sqrt6)`
for stable eq., `(d^(2)U)/(dx^(2)) gt 0`
`(d^(2)U)/(dx^(2)) = 12 x - 12`
`(d^(2)U)/(dx^(2)) gt 0` at `x = 1 + (1)/(sqrt6)`
& `(d^(2)U)/(dx^(2)) lt 0` at `x = 1 - (1)/(sqrt6)`