Consider The Function { F(x)=\frac{1}{x^2+4x+13} $}$.a) Give The Domain Of { F $}$ (in Interval Notation). { \square$}$b) Find The Critical Numbers Of { F $}$. { \square$}$ (Separate Multiple Answers
Analyzing the Function and Finding Critical Numbers
Understanding the Function
The given function is { f(x)=\frac{1}{x^2+4x+13} $}$. To analyze this function, we need to understand its behavior and properties. The function is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is a constant (1), and the denominator is a quadratic polynomial.
Domain of the Function
To find the domain of the function, we need to determine the values of x for which the function is defined. Since the function is a rational function, it is undefined when the denominator is equal to zero. Therefore, we need to find the values of x that make the denominator zero.
The denominator of the function is { x^2+4x+13 $}$. To find the values of x that make this expression equal to zero, we can set it equal to zero and solve for x:
{ x^2+4x+13=0 $}$
This is a quadratic equation, and we can solve it using the quadratic formula:
{ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} $}$
In this case, a = 1, b = 4, and c = 13. Plugging these values into the formula, we get:
{ x=\frac{-4\pm\sqrt{4^2-4(1)(13)}}{2(1)} $}$
Simplifying the expression under the square root, we get:
{ x=\frac{-4\pm\sqrt{16-52}}{2} $}$
{ x=\frac{-4\pm\sqrt{-36}}{2} $}$
Since the square root of a negative number is not a real number, this equation has no real solutions. Therefore, the denominator of the function is never equal to zero, and the function is defined for all real numbers.
The domain of the function is the set of all real numbers, which can be written in interval notation as:
{ (-\infty,\infty) $}$
Critical Numbers
To find the critical numbers of the function, we need to find the values of x for which the derivative of the function is equal to zero or undefined.
The derivative of the function can be found using the quotient rule:
{ f'(x)=\frac{(x2+4x+13)(0)-(1)(2x+4)}{(x2+4x+13)^2} $}$
Simplifying the expression, we get:
{ f'(x)=\frac{-2x-4}{(x2+4x+13)2} $}$
To find the critical numbers, we need to set the derivative equal to zero and solve for x:
{ \frac{-2x-4}{(x2+4x+13)2}=0 $}$
Since the denominator is never equal to zero, we can set the numerator equal to zero and solve for x:
{ -2x-4=0 $}$
Solving for x, we get:
{ x=-2 $}$
Therefore, the critical number of the function is x = -2.
Conclusion
In conclusion, the domain of the function { f(x)=\frac{1}{x^2+4x+13} $}$ is the set of all real numbers, which can be written in interval notation as { (-\infty,\infty) $}$. The critical number of the function is x = -2.
Discussion
The function { f(x)=\frac{1}{x^2+4x+13} $}$ is a rational function, and its domain is the set of all real numbers. The critical number of the function is x = -2, which means that the function has a local extremum at this point.
To find the local extremum, we need to evaluate the function at the critical point:
{ f(-2)=\frac{1}{(-2)^2+4(-2)+13} $}$
Simplifying the expression, we get:
{ f(-2)=\frac{1}{4-8+13} $}$
{ f(-2)=\frac{1}{9} $}$
Therefore, the local extremum of the function is { \frac{1}{9} $}$.
Final Answer
The final answer is:
- Domain: { (-\infty,\infty) $}$
- Critical number: { x=-2 $}$
Note: The final answer is in the format of a list, with each item corresponding to a part of the problem. The first item is the domain of the function, and the second item is the critical number of the function.
Q&A: Analyzing the Function and Finding Critical Numbers
Q: What is the domain of the function { f(x)=\frac{1}{x^2+4x+13} $}$?
A: The domain of the function is the set of all real numbers, which can be written in interval notation as { (-\infty,\infty) $}$. This is because the denominator of the function is never equal to zero, and the function is defined for all real numbers.
Q: How do you find the critical numbers of the function?
A: To find the critical numbers of the function, you need to find the values of x for which the derivative of the function is equal to zero or undefined. You can do this by setting the derivative equal to zero and solving for x, or by finding the values of x that make the derivative undefined.
Q: What is the critical number of the function { f(x)=\frac{1}{x^2+4x+13} $}$?
A: The critical number of the function is x = -2. This is because the derivative of the function is equal to zero at x = -2, which means that the function has a local extremum at this point.
Q: How do you find the local extremum of the function?
A: To find the local extremum of the function, you need to evaluate the function at the critical point. In this case, the local extremum of the function is { \frac{1}{9} $}$.
Q: What is the significance of the domain and critical numbers of the function?
A: The domain and critical numbers of the function are important because they help you understand the behavior of the function. The domain tells you where the function is defined, and the critical numbers tell you where the function has local extrema.
Q: How do you use the domain and critical numbers to analyze the function?
A: You can use the domain and critical numbers to analyze the function by evaluating the function at the critical points and determining the behavior of the function in different intervals.
Q: What are some common mistakes to avoid when analyzing the function?
A: Some common mistakes to avoid when analyzing the function include:
- Not checking the domain of the function
- Not finding the critical numbers of the function
- Not evaluating the function at the critical points
- Not determining the behavior of the function in different intervals
Q: How do you determine the behavior of the function in different intervals?
A: You can determine the behavior of the function in different intervals by evaluating the function at test points in each interval. If the function is increasing at a test point, then the function is increasing in that interval. If the function is decreasing at a test point, then the function is decreasing in that interval.
Q: What are some real-world applications of analyzing the function?
A: Some real-world applications of analyzing the function include:
- Modeling population growth
- Modeling economic systems
- Modeling physical systems
- Modeling biological systems
Q: How do you use the function to model real-world phenomena?
A: You can use the function to model real-world phenomena by substituting real-world data into the function and analyzing the resulting behavior. For example, you can use the function to model population growth by substituting population data into the function and analyzing the resulting behavior.
Q: What are some common challenges when analyzing the function?
A: Some common challenges when analyzing the function include:
- Finding the domain and critical numbers of the function
- Evaluating the function at the critical points
- Determining the behavior of the function in different intervals
- Using the function to model real-world phenomena
Q: How do you overcome these challenges?
A: You can overcome these challenges by:
- Using mathematical techniques to find the domain and critical numbers of the function
- Using numerical methods to evaluate the function at the critical points
- Using graphical methods to determine the behavior of the function in different intervals
- Using real-world data to model real-world phenomena
Q: What are some future directions for research in analyzing the function?
A: Some future directions for research in analyzing the function include:
- Developing new mathematical techniques for finding the domain and critical numbers of the function
- Developing new numerical methods for evaluating the function at the critical points
- Developing new graphical methods for determining the behavior of the function in different intervals
- Developing new applications for the function in real-world phenomena.