# Minimize integral

#### idontknow

If $$\displaystyle \int_{0}^{\pi } y^2 dx =1 \; , \; y(0)=y(\pi)=0$$ . Minimize $$\displaystyle I=\int_{0}^{\pi} y'^2 dx$$.
should I use partial integration and where ? I'm a beginner on these type of problems and the book gives not a single hint.

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#### idontknow

$$\displaystyle y=\pm \sqrt{2/ \pi } \sin(x)$$ ; $$\displaystyle \; y(0)=y(\pi) =0$$ (correct).
$$\displaystyle \; \; 2/ \pi \int_{0}^{\pi} \sin^2 (x) dx =2 / \pi \cdot \pi /2 =1$$ (correct).
$$\displaystyle \min I=2/ \pi \int_{0}^{\pi} \cos^2 (x) dx =1$$ (correct).

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#### SDK

$$\displaystyle y=\pm \sqrt{2/ \pi } \sin(x)$$ ; $$\displaystyle \; y(0)=y(\pi) =0$$ (correct).
$$\displaystyle \; \; 2/ \pi \int_{0}^{\pi} \sin^2 (x) dx =2 / \pi \cdot \pi /2 =1$$ (correct).
$$\displaystyle \min I=2/ \pi \int_{0}^{\pi} \cos^2 (x) dx =1$$ (correct).
This is a calculus of variations problem. You should take the functional derivative of $I$ and set it to zero to get the corresponding Euler-Lagrange equations. Where did you get that $y = \pm \sqrt{2/\pi} \sin (x)$?