Question 137
Define an order between two polynomials $f(x) \geq g(x)$ by comparing coefficients starting from the highest degree, that is, the following statement: if $f(x) = \sum_{i=0}^{n} a_i x^i$, $g(x) = \sum_{i=0}^{n} b_i x^i$ ($a_n, b_n$ may be $0$), and there exists $r$ such that $a_i = b_i$ for all $i \gt r$ and $a_r \gt b_r$, or $f(x) = g(x)$, then we say $f(x) \geq g(x)$.
Prove that if the leading coefficients of $f, g$ are positive, then $f(f(x)) + g(g(x)) \geq f(g(x)) + g(f(x))$.
Prove that if the leading coefficients of $f, g$ are positive, then $f(f(x)) + g(g(x)) \geq f(g(x)) + g(f(x))$.