Can $f(x)=ax^{2k-1}+...$ and $g(x)=bx^{2n}+...$ ($k,n \in \mathbb N$) have the same shape at some interval?

Question

When $k \in \mathbb N$ and $n \in \mathbb N$

$f(x)$ is a $2k-1$ degree polynomial

$g(x)$ is a $2n$ degree polynomial.

By 'having the same shape at some interval', I mean that when given proper $f(x)$ and $g(x)$ , the graphs of $f(x)$ and $g(x)$ are identical at some interval $[p, q]$.

Can the two functions have the same shape at some interval?

Answer 1

No, this will mean that $f(x)-g(x)=0$ will have infinitely many solutions on the interval $[p,q]$ which is impossible because this equation has at most $\max(2k-1, 2n)$ real roots

Answer 2

As polynomials, $f$ has $2k$ coefficients, and $g$ has $2n+1$ coefficients. Let $M=\max(2k, 2n+1)$; if $f(x) = g(x)$ at $M+1$ different points, then all their coefficients must be equal, which means $f=g$ everywhere. If they are equal on an interval, then they are equal at infinitely many points, and therefore equal everywhere.

If you take $f$ and $g$ of different degree, or you ensure at least one coefficient is different, then they can never agree on an interval.

Hope this helps!