Problem $3)$ Utilize the Rayleigh$-$Ritz Energy Method to estimate the critical buckling load $P_{cr}$ of the beam$-$

spring system shown below. In your solution utilize the following $1-$term expansion to approximate the beam's

deflection $w(x)=A_1{x}^3.$

Display your answer completely in terms of the given parameters $E,$I,$L,k$ and $P.$

Hint: Recall that the foreshortening work done by and axial compressive load $P$ is :

$W={\displaystyle \underset{0}{\overset{L}{}}}P\frac{w^{\text{'}}(x{)}^2}{2}dx$