Track / Stage 5 / 13 of 400 #613 of 1964

Problem 613

AIME late

Algebra Difficulty 5.0 Find the answer

$13 \cdot 16$ If $0<x<1$ , then among $x^{2}, x, \sqrt{x}, \frac{1}{x}$
(A) $\frac{1}{x}$ is the largest, $x^{2}$ is the smallest.
(B) $x$ is the largest, $\frac{1}{x}$ is the smallest.
(C) $x^{2}$ is the largest, $\sqrt{x}$ is the smallest.
(D) $x$ is the largest, $x^{2}$ is the smallest.
(China Junior High School Mathematics Correspondence Competition, 1987)

This was a multiple-choice question, but the options didn't survive into the source we have, so there is nothing here to pick from. Work it on paper and mark yourself against the solution below.

Official solution

[Solution] From the given condition $01 \text {. }$

Since $x-x^{2}=x(1-x)>0$ , we have $x>x^{2}$ . Also, $\sqrt{x}-x^{2}=\sqrt{x}(1-x \sqrt{x})>0$ , so $\sqrt{x}>x^{2}$ . Therefore, $\frac{1}{x}$ is the largest, and $x^{2}$ is the smallest. Hence, the answer is $(A)$ .

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