How many times can these two polynomial intersect with each other? Suppose $g(x)=a_1x+a_2x^2+...+a_kx^k$ and $f(x)=b_jx^j$ where $a_1,a_2...a_k&gt;0$ , $j\in \{1,2.....,k-1\}$, $b_j &gt;0$ and $x\geq0$.

Intuitively, I think they can have at most two intersections. Is that correct?

Well, the answer is it has two positive roots by Descartes' rule of signs. Thanks for your help!
The general answer to this question is given by Bezout's theorem. It states that two polynomials of degree $n$ and $m$ have precisely $n\cdot m$ intersection points provided that you


are over an algebraically closed field
count including multiplicities 
also count possible intersections at infinity.


This is overkill for your situation but when applied to polynomials over the reals it still means you can have at most $m\cdot n$ intersections. One can find examples where you actually have that many real intersections.
The general answer to this question is given by Bezout's theorem. It states that two polynomials of degree $n$ and $m$ have precisely $n\cdot m$ intersection points provided that you


are over an algebraically closed field
count including multiplicities 
also count possible intersections at infinity.


This is overkill for your situation but when applied to polynomials over the reals it still means you can have at most $m\cdot n$ intersections. One can find examples where you actually have that many real intersections.