Derive the Euler-Lagrange equation for the integral

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

where prime denotes differentiation with respect to $x$, and both $y$ and $y^{\prime}$ are specified at $x=a, b$.

Find $y(x)$ that extremises the integral

$$\int_{0}^{\pi}\left(y+\frac{1}{2} y^{2}-\frac{1}{2} y^{\prime \prime 2}\right) d x$$

subject to $y(0)=-1, y^{\prime}(0)=0, y(\pi)=\cosh \pi$ and $y^{\prime}(\pi)=\sinh \pi$.

Show that your solution is a global maximum. You may use the result that

$$\int_{0}^{\pi} \phi^{2}(x) d x \leqslant \int_{0}^{\pi} \phi^{\prime 2}(x) d x$$

for any (suitably differentiable) function $\phi$ which satisfies $\phi(0)=0$ and $\phi(\pi)=0$.