When f ‘(x) is negative, f(x) is
When the derivative of a function f(x), f'(x), is negative, it means that the function is decreasing
When the derivative of a function f(x), f'(x), is negative, it means that the function is decreasing.
Let’s understand this concept further:
– If f'(x) < 0 for all x in the domain, then the function f(x) is continuously decreasing. This means that as x increases, f(x) decreases.
- If f'(x) < 0 over a specific interval, it indicates that the function is decreasing on that interval, but it could be increasing or constant on other intervals.
To illustrate this, let's take an example:
Consider the function f(x) = x^2 - 3x. We will find the derivative and analyze the sign of f'(x) to understand how f(x) behaves.
1. Calculate f'(x):
To find the derivative of f(x), we apply the power rule and constant rule:
f'(x) = d/dx(x^2 - 3x)
= 2x - 3
2. Analyze the sign of f'(x):
To determine the intervals where f(x) is increasing or decreasing, we solve the inequality f'(x) < 0:
2x - 3 < 0
Solving the inequality:
2x < 3
x < 3/2
So, f'(x) is negative (less than zero) when x < 3/2.
3. Determine the behavior of f(x):
Since f'(x) < 0 when x < 3/2, we can conclude that f(x) is a decreasing function on the interval (−∞, 3/2). This means that as x increases from negative infinity (−∞) to 3/2, the values of f(x) decrease.
In summary, when the derivative of a function f'(x) is negative, such as in the example above, it indicates that the function f(x) is decreasing.
More Answers:
Understanding Calculus: Exploring the Fundamentals and Applications of DerivativesUnderstanding the Concept of Derivatives: Calculating Rates of Change Using Limits
Understanding the Positive Derivative: How it Indicates an Increasing Function