Exercises

1. Find the extremal curves $y = y(x)$ of the following constrained optimization problems, using the method of Lagrange multipliers:
   1. $\int_0^1\left(y'^{\,2}+x^2\right)\,dx$ , such that $\int_0^1 y^{\,2}\,dx = 2$ .
   2. $\int_0^\pi y'^{\,2}\,dx$ , such that $y(0)=y(\pi)=0$ , and $\int_0^\pi y^{\,2} \,dx=2$ .
   3. $\int_0^1y\,dx$ , such that $y(0)=y(1)=1$ , and $\int\sqrt{1+y'^{\,2}}\,dx=2\pi/3$ .
2. Suppose $P$ and $Q$ are two points lying in the $x$ -$y$ plane, which is orientated vertically such that $P$ is above $Q$ . Imagine there is a thin, flexible wire connecting the two points and lying entirely in the $x$ -$y$ plane. A frictionless bead travels down the wire, impelled by gravity alone. Show that the shape of the wire that results in the bead reaching the point $Q$ in the least amount of time is a cycloid, which takes the parametric form

   $$x(\theta) =k\left(\theta-\sin\theta\right),$$

   $$y(\theta) =k \left(1-\cos\theta\right),$$

   
   
   where $k$ is a constant.
3. Find the curve $y(x)$ , in the interval $0\leq x\leq p$ , which is of length $\pi$ , and maximizes
   $$\int_0^p y\,dx.$$