I am trying to show that there are infinitely many functions that minimize the integral.
$$ \int_0^2 \left( \left( y' \right)^2 \left(1 + y'\right)^2 \right) \, {\rm d} x$$ subject to $y(0) = 1$ and $y(2) = 0$.
(They are continuous functions with piecewise continuous first derivatives.)
Consider the two lines given by the graphs of $$ f(x)=1-x \\ g(x)=2-x $$ Now pick $0<s<1$ and consider $$ h_s(x)= \left\{ \begin{array}{lcl} f(x) & \text{if} & 0\leq x<s\\ 1-s & \text{if} & s\leq x<1+s\\ g(x) & \text{if} & 1+s\leq x \leq 2 \end{array} \right. $$ then it's easy to see that $J(h_s)=0$ ($J$ is the functional in question, and note that trivially $J\geq0$), so that $h_s$ are minimizers for all $0<s<1$.