Let $x$ be a real number between 0 and 1. Which of the following statements is/are correct? I. $x^2 > x^3$.…
Let $x$ be a real number between 0 and 1. Which of the following statements is/are correct?
I. $x^2 > x^3$.
II. $x > \sqrt{x}$.
Select the correct answer using the code given below:
I only
II only
Both I and II
Neither I nor II
Solution
For $0 < x < 1$: Statement I - $x^2 > x^3$ since dividing by $x^2$ (positive) gives $1 > x$, which is true. So I is correct. Statement II - $x > \sqrt{x}$ is false: for $0 x$ (e.g. $x=0.25$, $\sqrt{x}=0.5 > 0.25$). So II is incorrect. Only I is correct.