# Find the minimum of $F(y) = \int f(x,y(x)) dx$, w.r.t $y$, $f$ is given

find $\theta(x),$ such that the minimum value of

$F=S\int_0^l \left\{\frac{1}{2} k[\frac{d\theta(x)}{dx}]^2-\frac{1}{2}E^2 \cdot \triangle \epsilon \cdot \cos^2 \theta (x)\right\}dx$

is achieved.

Actually I have the answer from my book, but no reasoning is stated there, could any one give me a hand?

Note that $E, \triangle \epsilon, k, S, l$ are constants here, indicating some physics parameters.

Sorry for having made a typo in the previous post, I've changed it from $cos \theta(x)$ to $cos^2 \theta(x).$

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Firstly: the rule of this site is that rather than commanding one asks other people for advice. Secondly, I believe, you should restrict $S$ and $k$ (or at least $S k$) to be positive constants... –  Fabian Apr 12 '12 at 16:21
1.Thanks for your advice, I'm not familiar with the rules here, you can edit my post to a proper form, and I think asking for a proof is not that rare in this site. 2. $E, \triangle \epsilon, S, k, l$ are all positive constants. –  genxium Apr 12 '12 at 16:28

Hint: You can minimize the two terms simultaneously (they are not competing). The first term is minimized for $\theta'(x) =0$ which means ... The second term is when $- \Delta \epsilon \cos \theta$ assumes its minimum which means ...
but I'm quite confused by the notations, like $\int [\frac{d\theta(x)}{dx}]^2 dx$ seems a mass to me >_< –  genxium Apr 12 '12 at 16:25