Problem $05.$ For the following potential function

$U(r)=U_0(a{r}^4-b{r}^2)$

Find the equilibrium points $R_n$ in this potential, where $\frac{dU}{dr}=0$

For each equilibrium point, perform a taylor series expansion and determine whether

the point is stable or metastable. A stable equilibrium will experience a restoring force

which follow's hooke's law i$.$e$.U(R_n+r^{\text{'}})\frac{1}{2}k{r}^2$ for small $r^{\text{'}},$ leading to a restoring

force $F=-kx.$ An unstable equilibrium point will 'push the disturbance away' instead

of restoring, i$.$e$.U(R_n+r^{\text{'}})-\frac{1}{2}{k}^{\text{'}}{r}^2$ leading to $F=kx.$