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Finding the Largest Interval of Increasing Values for the Function f(x) = -x^3 + 4x + 3
In this article, we will explore the concept of increasing and decreasing functions, and how to find the largest interval of x-values where a given function is increasing. We will use the function f(x) = -x^3 + 4x + 3 and the given table of values to determine the largest interval of increasing values.
Understanding Increasing and Decreasing Functions
A function is said to be increasing on an interval if, for any two points x1 and x2 in the interval, f(x1) < f(x2) whenever x1 < x2. Conversely, a function is said to be decreasing on an interval if, for any two points x1 and x2 in the interval, f(x1) > f(x2) whenever x1 < x2.
Analyzing the Given Function
The given function is f(x) = -x^3 + 4x + 3. To find the largest interval of increasing values, we need to examine the behavior of the function as x varies.
Using the Table of Values
The table of values provides us with the following information:
| x | f(x) |
|---|---|
| -3 | 30 |
| -2 | 15 |
| -1 | 2 |
| 0 | 3 |
| 1 | 6 |
| 2 | 11 |
| 3 | 18 |
Finding the Largest Interval of Increasing Values
To find the largest interval of increasing values, we need to examine the table of values and identify the intervals where the function is increasing.
- For x = -3, f(x) = 30.
- For x = -2, f(x) = 15.
- For x = -1, f(x) = 2.
- For x = 0, f(x) = 3.
- For x = 1, f(x) = 6.
- For x = 2, f(x) = 11.
- For x = 3, f(x) = 18.
From the table, we can see that the function is increasing on the interval (-3, 3).
In conclusion, using the table of values for the function f(x) = -x^3 + 4x + 3, we have found that the largest interval of increasing values is (-3, 3). This means that for any x-value in the interval (-3, 3), the function f(x) is increasing.
Why is this Important?
Understanding the behavior of a function is crucial in many areas of mathematics and science. For example, in physics, the behavior of a function can describe the motion of an object. In economics, the behavior of a function can describe the relationship between two variables. In engineering, the behavior of a function can describe the performance of a system.
Real-World Applications
The concept of increasing and decreasing functions has many real-world applications. For example:
- In finance, the behavior of a function can describe the relationship between the price of a stock and the time of year.
- In medicine, the behavior of a function can describe the relationship between the concentration of a drug and the time it takes to reach a certain level.
- In engineering, the behavior of a function can describe the performance of a system, such as the relationship between the speed of a car and the time it takes to reach a certain speed.
In our previous article, we explored the concept of increasing and decreasing functions, and how to find the largest interval of x-values where a given function is increasing. We used the function f(x) = -x^3 + 4x + 3 and the given table of values to determine the largest interval of increasing values. In this article, we will answer some frequently asked questions about increasing and decreasing functions.
Q: What is the difference between an increasing and a decreasing function?
A: An increasing function is a function where, for any two points x1 and x2 in the interval, f(x1) < f(x2) whenever x1 < x2. Conversely, a decreasing function is a function where, for any two points x1 and x2 in the interval, f(x1) > f(x2) whenever x1 < x2.
Q: How do I determine if a function is increasing or decreasing?
A: To determine if a function is increasing or decreasing, you need to examine the behavior of the function as x varies. You can use the table of values, the graph of the function, or the derivative of the function to determine the behavior of the function.
Q: What is the largest interval of increasing values for the function f(x) = -x^3 + 4x + 3?
A: Using the table of values, we found that the largest interval of increasing values for the function f(x) = -x^3 + 4x + 3 is (-3, 3).
Q: How do I find the largest interval of increasing values for a given function?
A: To find the largest interval of increasing values for a given function, you need to examine the behavior of the function as x varies. You can use the table of values, the graph of the function, or the derivative of the function to determine the behavior of the function.
Q: What are some real-world applications of increasing and decreasing functions?
A: The concept of increasing and decreasing functions has many real-world applications. For example, in finance, the behavior of a function can describe the relationship between the price of a stock and the time of year. In medicine, the behavior of a function can describe the relationship between the concentration of a drug and the time it takes to reach a certain level. In engineering, the behavior of a function can describe the performance of a system, such as the relationship between the speed of a car and the time it takes to reach a certain speed.
Q: How do I use the derivative of a function to determine if it is increasing or decreasing?
A: To use the derivative of a function to determine if it is increasing or decreasing, you need to examine the sign of the derivative. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.
Q: What is the significance of the largest interval of increasing values for a given function?
A: The largest interval of increasing values for a given function is significant because it describes the behavior of the function as x varies. It can be used to make informed decisions in many areas of life, such as finance, medicine, and engineering.
In conclusion, the concept of increasing and decreasing functions is an important one in mathematics and science. By understanding the behavior of a function, we can gain valuable insights into the relationships between variables and make informed decisions in many areas of life.