Finding the amount of Horizontal Tangents

Question

I need help finding the number of points on the graph of h where there are horizontal tangent lines.

Here is a image link to the problem: https://i.imgur.com/ZQFbJOA.png

Thanks.

Answer 1

We need to find zeros of dh/dx or h'.

h(x) = f[g(x)] --> h'(x) = f'[g(x)]g'(x) = 0 --> 1) g'(x) = 0 or 2) f'[g(x)] = 0

1) g'(x) = 0 is true at three points from the given graph: x = -3, 0, 2.

2) f'[g(x)] = 0 --> g(x) = -2 or g(x) = 1 from the graph of f

2a) g(x) = -2 is true at two points x = -4, -2 from the g graph.

2b) g(x) = 1 is true at two points x = 0, 3.5ish from the g graph.

So there are 6 different points at x = -4, -3, -2, 0, 2, 3.5ish on the graph of h where there are horizontal tangent lines. Choice D is the correct answer.