Claim: Let $n\in \mathbb N$, and let $f:\mathbb R \to \mathbb R$ be such that its $n$-th derivative $f^{(n)}(x)>0, \ \forall x\in \mathbb R$, then $f$ has at most $n$ roots.
Context: The fundamental theorem of algebra states (when only the real numbers are concerned) that an $n$-th degree polynomial has at most $n$ real roots. Since an $n$-th degree polynomial can be characterized as a function $f:\mathbb R \to \mathbb R$ with constant $n$-th derivative, the fundamental theorem of algebra can be equivalently stated as:
Let $n\in \mathbb N$, and let $f:\mathbb R \to \mathbb R$ has constant $n$-th derivative, then $f$ has at most $n$ roots.
Analogy: Thus the claim I'm questioning can be interpreted as a version of the fundamental theorem of algebra with the condition that
the function's $n$-th derivative is constant
replaced by the condition that
the function's $n$-th derivative is strictly positive.
Special cases:
- for $n=1$ the claim follows from the fact that strictly increasing functions have at most one root;
- likewise for $n=2$ the claim follows from the fact that strictly increasing functions have at most two roots (strictly convex function is strictly negative between any two roots, thus there can not be a third root).
Does the claim hold for $n\geq 3$ as well?
As long as we're talking about distinct roots (not roots with multiplicity), then the answer is yes. We can deduce it from the following fact:
If $f(x)$ is differentiable and has at least $k$ distinct roots, then $f'(x)$ has at least $k-1$ distinct roots. (Proof: let $x_1<x_2<\cdots<x_k$ be roots of $f(x)$; then by Rolle's theorem, $f'(x)$ has a root in each of the disjoint intervals $(x_1,x_2)$, $(x_2,x_3)$, $\dots$, $(x_{k-1},x_k)$.)
The contrapositive of that fact is: If $f'(x)$ has at most $\ell$ distinct roots, then $f(x)$ has at most $\ell+1$ distinct roots. The OP's claim now follows by induction.
(Note that in the context of general functions, rather than only polynomials, the multiplicity of a root isn't a nice quantity; in particular it doesn't have to be an integer even if it is well defined.)