Under what condition on x is the tangent line to the curve horizontal? the curve has a horizontal tangent line when dy dx is

Short Answer dy/dx = 0

Remark.
The easiest way to show this is to draw it on a curve. When a line is horizontal it usually indicates a local minimum or maximum. 

For example y = (x - 3)(x - 1)(x + 1) 
y = x^3 - 3x^2 - x + 3

Differentiate
dy/dx = 3x^2 - 6x - 1

When you use the quadratic equation letting
a = 3
b = - 6
c = - 1
You get zeros at
x = 2.15
x = 0.15

Notice what happens. the graph shows you. You have a local max at x = - 0.15 and a local min at x = 2.15. See the graph on the right. 

The local horizontal max dy/dx is represented by a blue line
The local horizontal min dy/dx  is represented by a green line.