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.
