1957 AHSME Problems/Problem 29

Problem

The relation $x^2(x^2 - 1)\ge 0$ is true only for:

$\textbf{(A)}\ x \ge 1\qquad \textbf{(B)}\ - 1 \le x \le 1\qquad \textbf{(C)}\ x = 0,\, x = 1,\, x = - 1\qquad \\\textbf{(D)}\ x = 0,\, x\le-1,\, x\ge 1\qquad\textbf{(E)}\ x\ge 0$

Solution

To solve this problem, think about the graph of $f(x)=x^2(x^2-1)=x^2(x-1)(x+1)$ . The function equals zero only at the values $-1$ , $0$ , and $1$ . Because the function is a quartic polynomial with a positive leading coefficient, it will go to positive infinity as $x$ tends to either positive or negative infinity. Thus, when $x$ is greatly negative, the function will be positive, and so the given inequality will hold. As $x$ gradually becomes more positive, it will eventually equal $-1$ , when the function will equal zero. Thus, the $(x+1)$ term will switch to being positive, and so the whole function will become negative, where the inequality does not hold. $f(x)$ again reaches $0$ when $x=0$ , but here the $x^2$ term does not change sign, so the function stays negative afterwards. Finally, when $x=1$ , the function crosses the $x$ -axis as the $(x-1)$ term changes sign, and the function goes off to positive infinity. Thus, the function is positive (and thus the given inequality will hold) when $x \leq -1$ , $x=0$ , and $x \geq 1$ , which is answer choice $\fbox{\textbf{(D)}}$ .

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 28	Followed by Problem 30
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1957 AHSME Problems/Problem 29

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