4.9 Variational calculus

Problem: Let \( M=C_{1}[3 ; 5] \) - the class of the functions \( y(x) \), having a continuous derivative on the interval [3;5], and let \( I[y(x)]=\left(y^{\prime}\left(x_{0}\right)\right)^{3}+y^{3}\left(x_{0}\right) \), where \( x_{0} \in[3 ; 5] \). Select couple of functions \( y(x) \) and a point \( x_{0} \in[3 ; 5] \); then find the value of the functional for the taken function and point.