$9.$ Hooke's Law can be derived from the Taylor expansion of a I$-$D potential function in the vicinity of a local minimum.

$($a$)(5$ pts$)$ Write down the first three terms $($that is$,$ up to the quadratic term$)$ for the Taylor series of the potential function $\backslash ($ U$^\{\backslash$prime$\}($x$)\backslash )$ near a critical point, $\backslash ($ x$\_\{\backslash$ell$-\}\backslash )($b$)(5$ pts$)$ Use the Taylor oxries in part $($a$)$ to recover Hooke's Law, F $=-$kx$.$