we define the minimization problem :
$$\min J(x)=\int_{-1}^1x^2(t)[x'(t)-1]dt \quad \text{subject to }x(-1)=0 \quad \text{and} \quad x(1)=0$$
note that $0\leq J(x)$.
and if we consider $x^*(t)=0 \quad \text{for} t\in[-1,0]$ and $x^*(t)=t \quad \text{for} t\in[0,1]$.
it defines a solution de the problem.
But Why this problem doesn't have a solution in $C^1[-1,1]$ ?