Question

If \(c\) is any real number and \(0 < c < 1\), then:

• A. $\text{c} < \text{c}^2$
• B. $\text{c}^2 < \text{c}$
• C. $\text{c}^2 < \text{c}^3$
• D. $\text{c}^3 = 1$

Hint:

When a positive number is less than 1, multiplying it by itself makes it smaller.
Therefore, for \( 0 < c < 1 \):
\[ c > c^2 > c^3 > c^4 > \dots \]

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We analyze the behavior of powers of real numbers that lie strictly between 0 and 1.
Key Formula or Approach:
We can multiply an inequality by a positive real number without changing the direction of the inequality.

Step 2: Detailed Explanation:
We are given the inequality:
\[ 0 < c < 1 \]
Let us multiply all parts of this inequality by the positive real number \( c \):
\[ c \cdot 0 < c \cdot c < c \cdot 1 \]
Simplifying the terms:
\[ 0 < c^2 < c \]
Focusing on the relationship between \( c^2 \) and \( c \):
\[ c^2 < c \]
This shows that Option B is true, and Option A (\( c < c^2 \)) is false.
Now, let us multiply this resulting inequality \( c^2 < c \) by the positive number \( c \) again:
\[ c \cdot c^2 < c \cdot c \]
\[ c^3 < c^2 \]
This shows that Option C (\( c^2 < c^3 \)) is false.
Since \( c < 1 \), we have \( c^3 < 1^3 = 1 \), so Option D (\( c^3 = 1 \)) is false.

Step 3: Final Answer:
The correct inequality is $\text{c}^2 < \text{c}$ (Option B).