Using the First Derivative to Find the Range

Oct 13, 2025 by miodeya.com 128 views

In mathematics, functions are used to describe the relationship between variables. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this article, we will explore the domain and range of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ .

What is the Domain of a Function?

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x that can be plugged into the function without causing any problems. For example, if we have a function $f(x) = \frac{1}{x}$ , the domain of this function is all real numbers except for zero, because dividing by zero is undefined.

What is the Range of a Function?

The range of a function is the set of all possible output values. In other words, it is the set of all possible values of f(x) that can be obtained by plugging in different values of x. For example, if we have a function $f(x) = x^2$ , the range of this function is all non-negative real numbers, because the square of any real number is always non-negative.

Finding the Domain of the Function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$

To find the domain of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ , we need to determine the values of x for which the function is defined. In this case, the function is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of the function is all real numbers.

Finding the Range of the Function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$

To find the range of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ , we need to determine the set of all possible output values. In this case, the function is a polynomial, and polynomials can take on any real value. However, we need to consider the leading coefficient of the polynomial, which is $\frac{1}{4}$ . This means that the function will always be positive, because the leading coefficient is positive.

Using the First Derivative to Find the Range

To find the range of the function, we can use the first derivative to determine the behavior of the function. The first derivative of the function is given by:

f'(x) = \frac{3}{4}x^2 + \frac{12}{4}x + \frac{5}{4}

We can use the first derivative to determine the critical points of the function, which are the points where the function changes from increasing to decreasing or vice versa. To find the critical points, we need to set the first derivative equal to zero and solve for x:

\frac{3}{4}x^2 + \frac{12}{4}x + \frac{5}{4} = 0

Solving this equation, we get:

x = -\frac{4}{3} \text{ or } x = -1

These are the critical points of the function. To determine the behavior of the function, we can use the first derivative test. We can test the behavior of the function by plugging in values of x that are less than, equal to, and greater than the critical points.

Using the Second Derivative to Find the Range

To find the range of the function, we can use the second derivative to determine the concavity of the function. The second derivative of the function is given by:

f''(x) = \frac{3}{2}x + 3

We can use the second derivative to determine the concavity of the function. If the second derivative is positive, the function is concave up, and if the second derivative is negative, the function is concave down. To determine the concavity of the function, we can plug in values of x that are less than, equal to, and greater than the critical points.

Finding the Maximum and Minimum Values of the Function

To find the maximum and minimum values of the function, we can use the first and second derivatives. We can use the first derivative to find the critical points, and then use the second derivative to determine the concavity of the function. If the second derivative is positive at a critical point, the function is concave up at that point, and if the second derivative is negative at a critical point, the function is concave down at that point.

Conclusion

In this article, we have explored the domain and range of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ . We have found that the domain of the function is all real numbers, and the range of the function is all real numbers greater than or equal to zero. We have also used the first and second derivatives to find the maximum and minimum values of the function.

References

[1] Calculus, James Stewart, 8th edition
[2] Algebra, Michael Artin, 2nd edition
[3] Calculus, Michael Spivak, 4th edition

Glossary

Domain: The set of all possible input values for which the function is defined.
Range: The set of all possible output values.
First derivative: The derivative of a function with respect to one of its variables.
Second derivative: The derivative of a function's first derivative with respect to one of its variables.
Critical points: The points where the function changes from increasing to decreasing or vice versa.
Concavity: The shape of a function's graph, which can be concave up or concave down.
Q&A: Domain and Range of the Function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ ====================================================================

In this article, we will answer some common questions related to the domain and range of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ .

Q: What is the domain of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ ?

A: The domain of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ is all real numbers. This is because the function is a polynomial, and polynomials are defined for all real numbers.

Q: What is the range of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ ?

A: The range of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ is all real numbers greater than or equal to zero. This is because the leading coefficient of the polynomial is $\frac{1}{4}$ , which is positive.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, you need to determine the set of all possible input values and the set of all possible output values. For polynomials, the domain is all real numbers, and the range can be found by analyzing the leading coefficient and the degree of the polynomial.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In other words, the domain is the set of all possible x-values, and the range is the set of all possible f(x) values.

Q: Can a function have a domain that is not all real numbers?

A: Yes, a function can have a domain that is not all real numbers. For example, a function with a denominator of zero will have a domain that is all real numbers except for the value that makes the denominator zero.

Q: Can a function have a range that is not all real numbers?

A: Yes, a function can have a range that is not all real numbers. For example, a function that is always positive will have a range that is all non-negative real numbers.

Q: How do I use the first and second derivatives to find the domain and range of a function?

A: To use the first and second derivatives to find the domain and range of a function, you need to analyze the behavior of the function at its critical points. The first derivative can be used to find the critical points, and the second derivative can be used to determine the concavity of the function.

Q: What is the significance of the first and second derivatives in finding the domain and range of a function?

A: The first and second derivatives are used to analyze the behavior of a function at its critical points. The first derivative can be used to find the critical points, and the second derivative can be used to determine the concavity of the function. This information can be used to determine the domain and range of the function.

Q: Can a function have multiple domains and ranges?

A: No, a function cannot have multiple domains and ranges. The domain and range of a function are unique and are determined by the function's behavior.

Q: How do I determine the domain and range of a function with multiple variables?

A: To determine the domain and range of a function with multiple variables, you need to analyze the behavior of the function at its critical points. The first and second derivatives can be used to analyze the behavior of the function, and the domain and range can be determined based on this analysis.

Q: What is the relationship between the domain and range of a function and its graph?

A: The domain and range of a function are related to its graph. The domain is the set of all possible x-values, and the range is the set of all possible f(x) values. The graph of a function can be used to visualize the domain and range of the function.

Q: Can a function have a domain and range that are not related to its graph?

A: No, a function's domain and range are always related to its graph. The graph of a function can be used to visualize the domain and range of the function.

Q: How do I use the domain and range of a function to solve problems?

A: The domain and range of a function can be used to solve problems in a variety of ways. For example, you can use the domain and range to determine the possible values of a function, or to analyze the behavior of a function at its critical points.

Q: What are some common applications of the domain and range of a function?

A: The domain and range of a function have a wide range of applications in mathematics and other fields. Some common applications include:

Analyzing the behavior of a function at its critical points
Determining the possible values of a function
Visualizing the graph of a function
Solving problems in mathematics and other fields

Q: Can a function have a domain and range that are not related to its applications?

A: No, a function's domain and range are always related to its applications. The domain and range of a function can be used to analyze the behavior of the function and to determine its possible values, which are essential for solving problems in mathematics and other fields.