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Proving That a Function Is Decreasing When f(x) Is Less Than Zero

Proving That a Function Is Decreasing When f(x) Is Less Than Zero

January 07, 2025 Finance

Proving That a Function Is Decreasing When f'(x) Is Less Than Zero

Understanding the Concept of Function Decrease and the Role of the Derivative

When discussing the behavior of a function f(x), it's crucial to understand how its derivative can be used to determine whether the function is increasing or decreasing.

What Does It Mean for a Function to Be Decreasing?

A function f(x) is considered to be decreasing in a specified interval if, as the value of x increases, f(x) decreases. In other words, for any two points x1 and x2 within the interval where x1 , we have f(x1) > f(x2).

The Role of the Derivative in Determining Function Behavior

The derivative, denoted as f'(x), provides us with the rate of change of the function at any point x. When f'(x) , it indicates that the function is decreasing at that point. Let's delve deeper into this concept and explore the mathematical underpinnings.

Derivative and Increasing/Decreasing Functions

The relationship between the derivative and the behavior of a function can be succinctly described as follows:

When f'(x) > 0, the function is increasing. When f'(x) When f'(x) 0, the function has a critical point which could be a maximum, minimum, or an inflection point.

Mathematical Proof Using Limits

To rigorously prove that a function f(x) is decreasing when f'(x) in a given interval, we can use the definition of the derivative and the concept of limits. Let's consider a function f defined on the real line R, and let's look at a specific interval [a, b].

Interval and the Definition of Derivative

Consider the interval [a, b] and assume that for all x in this interval, f'(x) . We will use the definition of the derivative to demonstrate that f(x) is decreasing:

ouncer{f(x h) - f(x)}{h} for sufficiently small values of (h) (neither positive nor negative). Case 1: h > 0

When (h > 0), the denominator of the derivative is positive. Therefore, for the derivative to be less than zero, the numerator f(x h) - f(x) must be negative:

f(x h) - f(x)

or

f(x h)

This shows that as (x) increases by (h), the value of f(x) decreases, indicating that the function is decreasing in this interval. Case 2: h

When (h f(x h) - f(x) must be positive:

f(x h) - f(x) > 0

or

f(x h) > f(x)

Although the numerator is positive in this case, it still aligns with the overall decreasing nature of the function when considering the behavior of the function values: f(x) > f(x-h)

Both cases support the conclusion that if f'(x) at any point in the interval [a, b], the function is decreasing in that interval.

Conclusion

In summary, a mathematical function f(x) is decreasing in a given interval if, at each point in that interval, the derivative of the function is less than zero. Using the definition of the derivative and the concept of limits, we can rigorously prove that the function is decreasing in that interval. This fundamental property of derivatives is widely used in calculus to understand and analyze the behavior of functions and is crucial for many applications in mathematics, engineering, and the sciences.

Keywords

function decreasing derivative less than zero proving decreasing function

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