Question : Let $0<x<1$. Then the correct inequality is:
Option 1: $x<\sqrt{x}<x^{2}$
Option 2: $\sqrt{x}<x<x^{2}$
Option 3: $x^{2}<x<\sqrt{x}$
Option 4: $\sqrt{x}< x^{2}<x$
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Correct Answer: $x^{2}<x<\sqrt{x}$
Solution :
Given: $0 < x < 1$
For $0 < x < 1$, it's always true that $x^{2} < x$ because squaring a fraction between 0 and 1 makes it smaller.
Similarly, it's always true that $x < \sqrt{x}$ because taking the square root of a fraction between 0 and 1 makes it larger.
So, the correct inequality is $x^{2}<x<\sqrt{x}$.
Hence, the correct answer is $x^{2}<x<\sqrt{x}$
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