Finding the amount of tangents on a graph

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

The function is a composition, which I am not too familiar with. Any help is appreciated.

Thanks.

Answer 1

Remember the chain rule, and the answer falls out.

h(x)' = (f(g(x))' = f'(g(x))g'(x)

The horizontal tangents for h(x) are where h'(x) = 0

Thus, f'(g(x))g'(x) = 0, so f'(g(x) = 0 or g'(x) = 0

f has tangents at -2 and 1. g has tangents at -3, 0, and 2.

We immediately see tangents at the three g values since this is the inner function.

Then, we need to see where g(x) = -2 and 1.

g(x) = -2 when x = -4 and -2.

g(x) = 1 when x = 0 and 3.3

Note that 0 is repeated,

3 + 2 + 2 - 1 = 6