I am asked to show that the fuctions $e^x, \cos(x) \text{ and } x^2$ are linearly independent.
I wanted to use the Wronskian determinant in order to prove the above property. We have: $$W= \begin{vmatrix} x^2 & \cos(x) & e^x\\ 2x & -\sin(x) & e^x\\ 2 & -\cos(x) & e^x\end{vmatrix}\\ = e^x [ (x^2-4x+2)\cos(x)+(2-x^2)\sin(x)]$$
But this function clearly has zeros, which should imply linear dependence. Did I do or say something wrong ?