Given two functions $f,g: A \subset \mathbb{R} \rightarrow \mathbb{R}$ , $f,g \in C^1(A)$, I want to study the number of points of intersection of these two functions.
So I can take $h(x) := f(x)-g(x)$, $h \in C^1(A)$, and thus I want to study the roots of that function $h$. If I have two zeros, this means that, for mean value theorem, the derivative of $h$ is zero in some point between the roots.
My question is, there are conditions which allow us to say that given a root for the derivative $\alpha | h'(\alpha)=0$ there exist $\{x_1, x_2\} | h(x_1)=h(x_2)$?
Obviously there are counter examples to this ($h(x)=x^3$), but I intuitively suspect that if $h \in C^2, \alpha$ not a saddle point, then the result follow (intuitively because I'm thinking at the intermediate value theorem).