I'm trying to show that there are an infinite number of functions that minimize the integral: $\int_0^2[(y')^2*(1 + y')^2] dx$ subject to $y(0) = 1$ and $y(2) = 0$.
(They are continuous functions with piecewise continuous first derivatives.)
I'm trying to show that there are an infinite number of functions that minimize the integral: $\int_0^2[(y')^2*(1 + y')^2] dx$ 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 and consider $ h_s(x)= \left\{ \begin{array}{lcl} f(x) & \text{if} & 0\leq x 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.