If $0 < x < 1$, which of the following is greatest?

### Concept & Logic When $x$ is a fraction between $0$ and $1$, squaring the number makes it smaller ($x^2 < x$). Conversely, taking the reciprocal makes the number larger ($1/x > 1$). ### Step-by-Step Solution * **Given:** $0 < x < 1$. * **Analysis:** Let $x = 0.5$ (or $\frac{1}{2}$). * Option (a): $x = 0.5$ * Option (b): $x^2 = 0.25$ * Option (c): $\frac{1}{x} = \frac{1}{0.5} = 2$ * Option (d): $\frac{1}{x^2} = \frac{1}{0.25} = 4$ * **Comparison:** $4$ is clearly the greatest value. ### Exam Strategy & Shortcut Remember this rule: For $0 < x < 1$, the order of magnitude is $x^2 < x < 1 < \frac{1}{x} < \frac{1}{x^2}$. Any fraction between $0$ and $1$ squared will shrink, while its reciprocal will grow. ### Common Pitfall Students often think $x^2$ is the greatest because they are used to integers where squaring increases value. Always check the range of $x$ first. ### Final Answer Therefore, the correct answer is **$\\frac{1}{x^2}$**.