$[2]$ Using the variational formulation, prove the concept of equivalent spring constants for linear springs in $($i$)$ Parallel

and $($ii$)$ Series. $[60$pts$]$

i$)$ Parallel: $K_{eq}=k_1+k_2+k_3$

ii$)$ Series : $K_{eq}=\frac{k_1{k}_2{k}_3}{k_2{k}_3+k_1{k}_3+k_1{k}_2}$

$($Parallel$)$

$($Series$)$

$($Equivalent$)$