Track / Stage 5 / 233 of 400 #833 of 1964

Problem 833

AIME late

Algebra Difficulty 5.5 Prove it

10.2 Does there exist a function $h(x)$ that is decreasing on the interval $[0, \infty)$ such that the function $f(x)=\left(x^{2}-x+1\right) h(x)$ is increasing on the interval $[0, \infty) ?$

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Answer: does not exist.

Solution: If such a function existed, then it would be $f(1)>f(0)$ , i.e., $h(1)>h(0)$ , which is impossible due to the decreasing nature of the function $h(x)$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.