Question

If 0 < x < 1, which of the following has the greatest value?
a. x
b. ${{x}^{2}}$
c. ${{x}^{3}}$
d. $\dfrac{1}{x}$
e. $\dfrac{1}{{{x}^{2}}}$

Answer 1

Hint: In order to solve this question, we will consider a few values of x for 0 < x < 1 and then we will calculate $x,{{x}^{2}},{{x}^{3}},\dfrac{1}{x}$ and $\dfrac{1}{{{x}^{2}}}$ and from those values we will find which of them will show the greatest value.

Complete step-by-step answer:

In this question, we have been asked to find for, $x,{{x}^{2}},{{x}^{3}},\dfrac{1}{x}$ and $\dfrac{1}{{{x}^{2}}}$ which of them will have the greatest value. To solve this question, we will consider few values of x for 0 < x < 1 and check among $x,{{x}^{2}},{{x}^{3}},\dfrac{1}{x}$ and $\dfrac{1}{{{x}^{2}}}$ which gives the greatest value.
So, let us consider a few values of x for 0 < x < 1 as 0.25, 0.5 and 0.75.
So, for x = 0.25, we will get,
$\begin{align}
  & x=0.25 \\
 & {{x}^{2}}={{\left( 0.25 \right)}^{2}}=0.0625 \\
 & {{x}^{3}}={{\left( 0.25 \right)}^{3}}=0.015625 \\
 & \dfrac{1}{x}=\dfrac{1}{0.25}=4 \\
 & \dfrac{1}{{{x}^{2}}}=\dfrac{1}{{{\left( 0.25 \right)}^{2}}}=\dfrac{1}{0.0625}=16 \\
\end{align}$
From the above results, we can say that for x = 0.25, we get the greatest value for $\dfrac{1}{{{x}^{2}}}$.
Now, let us consider the values of $x,{{x}^{2}},{{x}^{3}},\dfrac{1}{x}$ and $\dfrac{1}{{{x}^{2}}}$ for x = 0.5. So, we will get,
$\begin{align}
  & x=0.5 \\
 & {{x}^{2}}={{\left( 0.5 \right)}^{2}}=0.25 \\
 & {{x}^{3}}={{\left( 0.5 \right)}^{3}}=0.125 \\
 & \dfrac{1}{x}=\dfrac{1}{0.5}=2 \\
 & \dfrac{1}{{{x}^{2}}}=\dfrac{1}{{{\left( 0.5 \right)}^{2}}}=\dfrac{1}{0.25}=4 \\
\end{align}$
From the above results, we again get that $\dfrac{1}{{{x}^{2}}}$ has the greatest value.
Now, we will check for x = 0.75. So, we will get,
$\begin{align}
  & x=0.75 \\
 & {{x}^{2}}={{\left( 0.75 \right)}^{2}}=0.5625 \\
 & {{x}^{3}}={{\left( 0.75 \right)}^{3}}=0.421875 \\
 & \dfrac{1}{x}=\dfrac{1}{0.75}=1.33 \\
 & \dfrac{1}{{{x}^{2}}}=\dfrac{1}{{{\left( 0.75 \right)}^{2}}}=\dfrac{1}{0.5625}=1.778 \\
\end{align}$
From the above results, we again get the greatest value for $\dfrac{1}{{{x}^{2}}}$.
So, we can say that for, 0 < x < 1, $\dfrac{1}{{{x}^{2}}}$ has the greatest value.
Hence, option (e) is the correct answer.

Note: We also have to remember this property of real numbers, that, for 0 < x < 1, the more the value of the power of x is, the lesser is the value of x. Similarly, the less the value of the power of x is, greater is the value of x.