Estimation of ground state energy from wave function
The variational method uses a linear combination of functions as an estimate of the true wavefunction. It works by adjusting coefficients in the linear combination to minimize the energy of the trial wavefunction. This is the best estimate of the true wavefunction using the restricted basis set. The variational principle states that adding more (new) basis functions to the linear combination always improves the estimate. 

In your problem (or according to your instructions), you don't really optimize anything (except for the last comparison), because the normalization condition fixes the value of the $c_i$ coefficients, but you can estimate which of the two functions appears most like the true wavefunction, as this should return a minimum energy. 

The ground state solution has the form $$\psi \propto sin(\pi x)$$
when x is in units of the box length, with bounds at $x=0$ and $x=1$ (and null wavefunction outside these bounds). This function has nodes at $x=0$ and $x=1$ and is concave with a single maximum at $x=\frac12$. By evaluating the functions at the bounds, in the midpoint and at one or more other points you should get an idea about the shape of each function and thereby which are better. It would of course be easier to plot them. Consider to the extent possible symmetry and whether bounds are satisfied (which they must for the solution to be at all physically reasonable - remember that the energy blows up if the wavefunction samples points out of bounds. 

The last comparison is a "trick question". There you have to take into account the variational principle which is mentioned in the first paragraph of this answer.