[30pts] Determine the extreinals for the functional associated With 1 y) = f [(gf'PZoayyF 2/3 y'ldsc U where 0: and [3 are constants, in each of the following situations: (a) The end point conditions y(0)=0 and y(1)=1 are preassigned. (h) Only the end condition y(0)=0 is preassigned. (c) Only the end condition y(1)=1 is preassigned. (d) No end conditions are preassigned, Theorem 3.5.1 Let I be a functional of the form dx I(2) = F a(t), ti dt (t), t dt, where F(x, x', t) is a function with continuous first and second partial derivatives with respect to (x, x', t) and x E C' [ti, tf] such that x(ti) = xi and x(t; ) = xf. If I has an extremum at X*, then xx satisfies the Euler-Lagrange equation. OF dx * d OF dx * * * (t ) , ax' =0, te [ti,tf]. dt (t ) , t dt (t), t) (3.9) dt X * (t)