Question 128
Let $P(x)$ and $Q(x)$ be polynomials with non-negative real coefficients, and let $P'(x)$ denote the derivative of $P(x)$. Suppose that we have $P(0) = Q(0) = 0$ and $Q(1) \le 1 \le P'(0)$.
(1) Prove that $0 \le Q(x) \le x \le P(x)$ for all $0 \le x \le 1$.
(2) Prove that $P(Q(x)) \le Q(P(x))$ for all $0 \le x \le 1$.
It is *not* necessary to study the conditions for equality.
(1) Prove that $0 \le Q(x) \le x \le P(x)$ for all $0 \le x \le 1$.
(2) Prove that $P(Q(x)) \le Q(P(x))$ for all $0 \le x \le 1$.
It is *not* necessary to study the conditions for equality.