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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.)

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    piecewise C^inf functions2012-12-02

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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.

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    @BuddyHolly: Sure, just derive the E-L equations and plug in. You'll get two possible general candidates. One being (arbitary) affine functions and the other some special affine functions. Discard some candidates by evaluating the functional and conclude that the only restriction is that the minimizers have $y'=0,-1$. (By the way you really should put the E-L restriction on the question)2012-12-02