Sign Of H ′ ( X ) H ( X ) − F ′ ( X ) F ( X ) H'(x)h(x) - F'(x) F(x) H ′ ( X ) H ( X ) − F ′ ( X ) F ( X )

Apr 20, 2025 by ADMIN 107 views

**Sign of $h'(x)h(x) - f'(x) f(x)$: A Comprehensive Analysis**

Introduction

In this article, we will delve into the analysis of the expression $h'(x)h(x) - f'(x) f(x)$ , where $f(x)$ and $h(x)$ are positive functions with positive derivatives. We will explore the conditions under which this expression is positive or negative, and provide a detailed explanation of the underlying mathematical concepts.

Given Conditions

We are given that $\xi > 0$ is a positive constant, and $x \in (0, 1/\xi)$ . Additionally, we know that $f(x) > 0$ and $h(x) > 0$ for all $x$ in the given interval. Furthermore, the derivatives of $f(x)$ and $h(x)$ are also positive, i.e., $f'(x) > 0$ and $h'(x) > 0$ . Lastly, we are given that $f(x) = \xi x h(x)$ .

Derivative of $f(x)$

To begin our analysis, let's find the derivative of $f(x)$ using the product rule:

f'(x) = \frac{d}{dx} (\xi x h(x))

f'(x) = \xi h(x) + \xi x h'(x)

Expression of Interest

Now, let's examine the expression $h'(x)h(x) - f'(x) f(x)$ . We can substitute the expression for $f'(x)$ that we derived earlier:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) - (\xi h(x) + \xi x h'(x)) \xi x h(x)

Simplifying the Expression

Let's simplify the expression by expanding the product:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) - \xi^2 x^2 h(x)^2 - \xi^2 x^3 h'(x) h(x)

Rearranging Terms

We can rearrange the terms to get:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) - \xi^2 x^2 h(x)^2 (1 + \xi x)

Factoring Out Common Terms

We can factor out common terms to get:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) (1 - \xi^2 x^2 (1 + \xi x))

Sign of the Expression

Now, let's analyze the sign of the expression. We know that $h'(x) > 0$ and $h(x) > 0$ , so the first term is positive. The second term is a product of two factors, and we can analyze the sign of each factor separately.

First Factor

The first factor is $1 - \xi^2 x^2 (1 + \xi x)$ . We can analyze the sign of this factor by considering the following cases:

If $\xi x < 1$ , then $1 + \xi x < 2$ , and the factor is positive.
If $\xi x > $, then $1 + \xi x > 2$ , and the factor is negative.

Second Factor

The second factor is $h'(x)h(x)$ . We know that $h'(x) > 0$ and $h(x) > 0$ , so this factor is always positive.

Combining the Factors

Now, let's combine the factors to determine the sign of the expression. We have:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) (1 - \xi^2 x^2 (1 + \xi x))

If $\xi x < 1$ , then the first factor is positive, and the expression is positive. If $\xi x > 1$ , then the first factor is negative, and the expression is negative.

Conclusion

In conclusion, the sign of the expression $h'(x)h(x) - f'(x) f(x)$ depends on the value of $\xi x$ . If $\xi x < 1$ , then the expression is positive. If $\xi x > 1$ , then the expression is negative.

References

[1] [Book Title], [Author], [Publisher], [Year]
[2] [Book Title], [Author], [Publisher], [Year]

Appendix

This section contains additional information that may be useful for readers who want to delve deeper into the topic.

Derivative of $h(x)$

To find the derivative of $h(x)$ , we can use the product rule:

h'(x) = \frac{d}{dx} (\xi x h(x))

h'(x) = \xi h(x) + \xi x h'(x)

Expression of Interest

We can substitute the expression for $h'(x)$ that we derived earlier:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) - (\xi h(x) + \xi x h'(x)) \xi x h(x)

Simplifying the Expression

We can simplify the expression by expanding the product:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) - \xi^2 x^2 h(x)^2 - \xi^2 x^3 h'(x) h(x)

Rearranging Terms

We can rearrange the terms to get:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) - \xi^2 x^2 h(x)^2 (1 + \xi x)

Factoring Out Common Terms

We can factor out common terms to get:

h'(x)h(x) - f'(x) f(x) = h'(x)h(x) (1 - \xi^2 x^2 (1 + \xi x))$<br/> **Q&A: Sign of $h'(x)h(x) - f'(x) f(x)$** =============================================

Q: What is the significance of the expression $h'(x)h(x) - f'(x) f(x)$ ?

A: The expression $h'(x)h(x) - f'(x) f(x)$ is significant because it represents a difference between two quantities, $h'(x)h(x)$ and $f'(x) f(x)$ . Understanding the sign of this expression is crucial in various mathematical and real-world applications.

Q: What are the given conditions for the functions $f(x)$ and $h(x)$ ?

A: The given conditions are that $\xi > 0$ is a positive constant, and $x \in (0, 1/\xi)$ . Additionally, we know that $f(x) > 0$ and $h(x) > 0$ for all $x$ in the given interval. Furthermore, the derivatives of $f(x)$ and $h(x)$ are also positive, i.e., $f'(x) > 0$ and $h'(x) > 0$ . Lastly, we are given that $f(x) = \xi x h(x)$ .

Q: How do we find the derivative of $f(x)$ ?

A: To find the derivative of $f(x)$ , we use the product rule:

Introduction

Given Conditions

Derivative of f(x)f(x)f(x)

Expression of Interest

Simplifying the Expression

Rearranging Terms

Factoring Out Common Terms

Sign of the Expression

First Factor

Second Factor

Combining the Factors

Conclusion

References

Appendix

Derivative of h(x)h(x)h(x)

Expression of Interest

Simplifying the Expression

Rearranging Terms

Factoring Out Common Terms

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Derivative of $f(x)$

Derivative of $h(x)$

Q: What is the significance of the expression $h'(x)h(x) - f'(x) f(x)$ ?

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