Problem 4 If \(0... [FREE SOLUTION]

Chapter 9: Problem 4

If \(0

Short Answer

Expert verified

(D) I and II only

Step by step solution

Understanding the Problem

We have three inequalities involving the variable \(x\), and we need to determine which inequalities hold true when \(0<x<1\). Our goal is to assess each statement (I, II, III) to determine the correct options.

Evaluating Statement I: \(x^2 < x\)

Since \(0 < x < 1\), when we square \(x\), the value becomes smaller because squaring a fraction yields a smaller fraction. So, \(x^2 < x\) holds true under this condition.

Evaluating Statement II: \(x < \frac{1}{x^2}\)

Consider the expression \(\frac{1}{x^2}\). Since \(0 < x < 1\), \(x^2 < 1\), making \(\frac{1}{x^2} > 1\). Therefore, \(x < \frac{1}{x^2}\) holds true because \(x\) is between 0 and 1.

Evaluating Statement III: \(\sqrt{x} < x\)

The square root of a number between 0 and 1 results in a larger number closer to 1, making \(\sqrt{x} > x\). Therefore, Statement III is false for \(0 < x < 1\).

Conclusion

Statements I and II are true, while Statement III is false. Based on the analysis, the correct answer is the option that includes Statements I and II.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Analysis

Inequality analysis involves understanding and comparing the sizes of different mathematical expressions based on given conditions. In the context of our problem, we consider inequalities involving a variable, \(x\), which lies in the range \(0
Let's explore the three statements:

For Statement I, \(x^2 < x\), squaring \(x\) results in a smaller number than \(x\) itself, since the square of a fraction less than one is even smaller. Thus, this inequality holds true.
For Statement II, \(x < \frac{1}{x^2}\), the term \(\frac{1}{x^2}\) becomes larger than one as \(x\) becomes smaller. Therefore, \(x\) is indeed less than \(\frac{1}{x^2}\), which makes this inequality true.
Lastly, Statement III, \(\sqrt{x} < x\), is false because the square root of \(x\) is always greater than \(x\) for \(0 < x < 1\), due to the property that square rooting brings the number closer to one.

By breaking down these inequalities, we can conclude that only Statements I and II are true for \(0

Mathematical Reasoning

Mathematical reasoning is essential for evaluating the truth of various propositions under specific conditions. It involves logical thought processes that allow us to deduce results from given premises. In our exercise, we have to carefully examine the expressions to determine if each inequality is true.

We apply reasoning to:

Statement I by recognizing that multiplying a fraction less than one by itself results in an even smaller fraction, essentially because \(x^2\) will always be less than \(x\) when \(0
Statement II by deducing that since \(x^2\) involves a fraction that reduces in size, \(\frac{1}{x^2}\) becomes larger as \(x^2\) gets nearer to zero, thus making \(x\) smaller comparatively.
Statement III by assessing how the square root function behaves, noting that for fractions, it yields a larger value closer to one compared to the original number, hence \(\sqrt{x} > x\) for this range.

This analysis and reasoning provides insight into why Statements I and II are valid, while III is not under the constraint that \(0

Logical Deduction

Logical deduction is a structured approach to reach conclusions based on initial conditions and known truths. It is about methodically processing the given information to evaluate statements.

In our problem, logical deduction can be seen in the systematic verification of inequalities:

For Statement I, logic is applied by recognizing that squaring a number between 0 and 1 results in a decrease in magnitude, confirming the inequality \(x^2 < x\).
For Statement II, deduction involves understanding the reciprocal operation, \(\frac{1}{x^2}\), contrasting \(x\) with its inverse squared, and recognizing that the outcome becomes larger than 1 due to how fractions behave when inverted.
With Statement III, logical deduction helps us discard the proposition as false. By realizing that \(\sqrt{x}\) increases the number's magnitude towards one, it contradicts the given inequality \(\sqrt{x} < x\).

Using logical deduction, we eliminate inability and solidify the truth behind Statements I and II being applicable, while III proves incorrect for \(0

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91影视

If \(0

Short Answer

Step by step solution

Understanding the Problem

Evaluating Statement I: \(x^2 < x\)

Evaluating Statement II: \(x < \frac{1}{x^2}\)

Evaluating Statement III: \(\sqrt{x} < x\)

Conclusion

Key Concepts

Inequality Analysis

Mathematical Reasoning

Logical Deduction

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