Here is a GIF demonstrating the tangent line as a point traverses a curve: https://s3-us-west-2.amazonaws.com/courses-images/wp-content...

The "pencil" (the short line representing the tangent) is green when positive, black when zero, and red when negative. Note that when it is black/zero, the "pencil" line is flat. This occurs at both local maxima and minima.