= Solution
The function $u_*(x)=x^{1/4}$ has <weak derivative> $u_*'(x)=\tfrac14x^{-3/4}\in L^1(0,1)$ and the required endpoint values, so it belongs to the admissible <Sobolev space>. Its fourth power is $x$, and hence
$$
\boxed{I_0(u_*)=0.}
$$
Since the integrand $(u^4-x)^2$ is nonnegative, zero is a global lower bound. Thus \b[$u_*$ is a <global minimizer>]. Its <derivative> does not belong to $L^{4/3}$, since its $4/3$ power is proportional to $x^{-1}$; positive regularization therefore needs different competitors.
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