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.

Lets use the letter U as your curve: U

One way to imagine the the horizontal tangent is that it's where the curve stops going down (negative slope) and starts going up (positive slope). Well, that is also basically the definition of a local minima, and one of those minima is also the global minima.

Try holding your mouse* upside-down on the surface of your desk to get the same effect. The point at which the mouse meets the desk will always be the point where the tangent of the curve is horizontal.

*Assuming your mouse is curved on top.

This was my "ah-ha" moment as well. I always knew that derivative of the lowest point is 0 but I didn't really understand what it meant. Take any arbitrary curve. Tangent is a line that touches a single point on the curve. Draw a tangent aka pencil touching the lowest point of the curve. It's derivative ie dy/dx apply the formulae to the curve's formulae. It will always be 0 which means it will have no "slope" ie parallel to the x axis

If the curve was inclined at that point, it would have to be lower either on the left or the right of it.

Just draw any curve in a piece of paper, you'll see it happening.

It might help to think of a specific curve. An example would be a simple parabola: f(x) = x^2. At the bottom of the curve (x = 0), the slope (aka tangent, aka derivative) is 0, or a horizontal line.