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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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Do you mean $C^1$ functions or piecewise $C^1$ functions? (if you need continous first derivatives I don't think this has minimizers) –  Jose27 Dec 2 '12 at 2:08
    
piecewise C^inf functions –  Buddy Holly Dec 2 '12 at 2:10

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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 $0<s<1$.

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The setup to use is the Euler Lagrange equation. Can you solve it using this? –  Buddy Holly Dec 2 '12 at 6:37
    
@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) –  Jose27 Dec 2 '12 at 16:27

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