Why can't the wave function and it's derivative be zero at the same point?

Question

This was discussed before proving the node theorem.

Answer 1

The 1D TISE is a second-order homogeneous ODE. The complete solution span a 2-dimensional vector space parametrized by 2 constants. It is not hard to see that if a solution and its derivative vanish in the same point, it must be the zero-solution. The zero-solution is unacceptable as a wave-function.