When proving this, my textbook first writes down the determinant, and then states:
This will be tedious to compute directly, and we only need to know that the Wronskian is nonzero at a single point, so we evaluate it at $x = 0$
Why is this?
I recall that there's a similar theorem that states if these functions are solutions to a differential equation
then they are linearly independent $\iff$ Wronskian = 0 at one point in the interval $\iff$ Wronskian = 0 at all points in the interval.
However, the problem in the header does not state that they are solutions to any differential equation.