Question

What does it mean about a point if the tangent line has a slope of 0?

Answer 1

A line with a slope of 0 is simply a horizontal line

Explanation:

As I said, the tangent is horizontal at that point. It has also meaning in terms of maxima and minima for the function. Since the tangent is horizontal at the point, this point is a good candidate for local minimum or minimum at the point (a critical point) . See for example the following local minimum `(0, 1)`:

graph{x^2+1 [-10, 10, -5, 5]}

where the tangent `y=1` is horizontal (i.e. has slope 0) at `(0, 1)`.

However, a `0` slope doesn't necessarily means that the point is maximum or minimum, as the following example shows you:

graph{x^3+1 [-10, 10, -5, 5]}

Where the slope is `0` at `(0,1)` but `(0,1)` is not neither maximum or minimum