In this exercise the theory developed in section 3.3.1 is extended. The function $F(z)$ has a continuous second derivative and the functional $S$ is defined by the integral
$$
S[y]=\int_a^b d x F\left(y^{\prime}\right)
$$
(a) Show that
$$
S[y+\epsilon h]-S[y]=\epsilon \int_a^b d x \frac{d F}{d y^{\prime}} h^{\prime}(x)+\frac{1}{2} \epsilon^2 \int_a^b d x \frac{d^2 F}{d y^{\prime 2}} h^{\prime}(x)^2+O\left(\epsilon^3\right),
$$
where $h(a)=h(b)=0$.
(b) Show that if $y(x)$ is chosen to make $d F / d y^{\prime}$ constant then the functional is stationary.
(c) Deduce that this stationary path makes the functional either a maximum or a minimum, provided $F^{\prime \prime}\left(y^{\prime}\right) \neq 0$.