Consider a functional

$$I=\int_{a}^{b} F\left(x, y, y^{\prime}\right) d x$$

where $F$ is smooth in all its arguments, $y(x)$ is a $C^{1}$ function and $y^{\prime}=\frac{d y}{d x}$. Consider the function $y(x)+h(x)$ where $h(x)$ is a small $C^{1}$ function which vanishes at $a$ and $b$. Obtain formulae for the first and second variations of $I$ about the function $y(x)$. Derive the Euler-Lagrange equation from the first variation, and state its variational interpretation.

Suppose now that

$$I=\int_{0}^{1}\left(y^{\prime 2}-1\right)^{2} d x$$

where $y(0)=0$ and $y(1)=\beta$. Find the Euler-Lagrange equation and the formula for the second variation of $I$. Show that the function $y(x)=\beta x$ makes $I$ stationary, and that it is a (local) minimizer if $\beta>\frac{1}{\sqrt{3}}$.

Show that when $\beta=0$, the function $y(x)=0$ is not a minimizer of $I$.