A particle located in a one-dimensional potential field has its potential energy function as U(x)= ax4−bx2  ,where a and b are positive constants. The position of equilibrium x corresponds to

The position of equilibrium corresponds to F(x) = 0Since F(x) =  −dU(x)dxSo F(x) = −ddx(ax4−bx2)or F(x) =  4ax5−2bx3For equilibrium F(x) = 0 , therefore4ax5−2bx3=0 ⇒  x= ±2abd2U(x)dx2=  −20ax6+6bx4Putting x = ±2ab gives d2U(x)dx2  as negativeSo U is maximum. Hence, it is position of unstable equilibrium