A6.* In Thereom 1 of the notes (the Euler-Lagrange equation), we require that y(x) = C...

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A6.* In Thereom 1 of the notes (the Euler-Lagrange equation), we require that y(x) = C [x0, x1] rather than y(x) = C [x0, x1]. (a) Show that the Euler-Lagrange equation is in fact a second-order differential equation. (b) The functional 1 J(y) = [ y (2x y') dx with boundary conditions y(-1) = 0 and y(1) = 1, has an extremal - 0 if 1x0, y(x) = x if 0 < x < 1. Find the value of x at which the extremal y has no second derivative (hint: what is the definition of derivative?) (c) Show that the Euler-Lagrange equation is still satisfied for the extremal. A6.* In Thereom 1 of the notes (the Euler-Lagrange equation), we require that y(x) = C [x0, x1] rather than y(x) = C [x0, x1]. (a) Show that the Euler-Lagrange equation is in fact a second-order differential equation. (b) The functional 1 J(y) = [ y (2x y') dx with boundary conditions y(-1) = 0 and y(1) = 1, has an extremal - 0 if 1x0, y(x) = x if 0 < x < 1. Find the value of x at which the extremal y has no second derivative (hint: what is the definition of derivative?) (c) Show that the Euler-Lagrange equation is still satisfied for the extremal.