Title: Existence of a bad local minimum of neural networks with general smooth activation functions. Abstract: Understanding the loss surface of neural networks is essential to the understanding of deep learning. However, the existence of a bad local minimum has not yet been fully identified. We investigate the existence of a bad local minimum of the $2$-layer and $3$-layer neural networks with general smooth activation functions. We provide constructive proof using the algebraic nature of the activation functions. We show this for realistic settings where the data $(X,Y)$ have a positive measure. We hope that such results give theoretical foundations for studies related to local minima and loss surfaces.