Finding the Critical Number

Oct 13, 2025 by talynt.org 117 views

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Introduction

In calculus, a critical number is a point in the domain of a function where the derivative of the function is equal to zero or undefined. Critical numbers are important in optimization problems, as they often correspond to maximum or minimum values of the function. In this article, we will find the critical number of the function $f(x)=(6x+9)e^{3x}$ .

The Derivative of $f(x)$

To find the critical number of $f(x)$ , we first need to find its derivative. We can use the product rule of differentiation, which states that if $f(x)=u(x)v(x)$ , then $f'(x)=u'(x)v(x)+u(x)v'(x)$ . In this case, we have $u(x)=6x+9$ and $v(x)=e^{3x}$ .

import sympy as sp
x = sp.symbols('x')
f = (6*x + 9)sp.exp(3x)
f_prime = sp.diff(f, x)

Using the product rule, we get:

f'(x)=(6x+9)(3e^{3x})+e^{3x}(6)

Simplifying the expression, we get:

f'(x)=18xe^{3x}+27e^{3x}+6e^{3x}

f'(x)=18xe^{3x}+33e^{3x}

f'(x)=e^{3x}(18x+33)

Finding the Critical Number

Now that we have the derivative of $f(x)$ , we can find the critical number by setting the derivative equal to zero and solving for $x$ .

e^{3x}(18x+33)=0

Since $e^{3x}$ is never equal to zero, we can divide both sides of the equation by $e^{3x}$ to get:

18x+33=0

Subtracting 33 from both sides, we get:

18x=-33

Dividing both sides by 18, we get:

x=-\frac{33}{18}

x=-\frac{11}{6}

Conclusion

In this article, we found the critical number of the function $f(x)=(6x+9)e^{3x}$ by first finding its derivative using the product rule of differentiation. We then set the derivative equal to zero and solved for $x$ to find the critical number. The critical number is $x=-\frac{11}{6}$ .

Final Answer

The final answer is $\boxed{-\frac{11}{6}}$ .

References

Note: The references provided are for general calculus textbooks and are not specific to the topic of critical numbers.

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Introduction

In our previous article, we found the critical number of the function $f(x)=(6x+9)e^{3x}$ by first finding its derivative using the product rule of differentiation. In this article, we will answer some frequently asked questions about critical numbers and derivatives.

Q: What is a critical number?

A: A critical number is a point in the domain of a function where the derivative of the function is equal to zero or undefined. Critical numbers are important in optimization problems, as they often correspond to maximum or minimum values of the function.

Q: How do I find the critical number of a function?

A: To find the critical number of a function, you need to first find its derivative using the rules of differentiation. Then, set the derivative equal to zero and solve for the variable. The point where the derivative is equal to zero is the critical number.

Q: What is the product rule of differentiation?

A: The product rule of differentiation is a rule that allows you to find the derivative of a product of two functions. If $f(x)=u(x)v(x)$ , then $f'(x)=u'(x)v(x)+u(x)v'(x)$ .

Q: How do I use the product rule to find the derivative of a function?

A: To use the product rule to find the derivative of a function, you need to identify the two functions that are being multiplied together. Then, find the derivatives of each function separately. Finally, plug these derivatives into the product rule formula and simplify.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation is a rule that allows you to find the derivative of a composite function. If $f(x)=g(h(x))$ , then $f'(x)=g'(h(x))\cdot h'(x)$ .

Q: How do I use the chain rule to find the derivative of a function?

A: To use the chain rule to find the derivative of a function, you need to identify the outer function and the inner function. Then, find the derivatives of each function separately. Finally, plug these derivatives into the chain rule formula and simplify.

Q: What is the relationship between critical numbers and optimization problems?

A: Critical numbers are important in optimization problems, as they often correspond to maximum or minimum values of the function. By finding the critical numbers of a function, you can determine the maximum or minimum value of the function.

Q: How do I use critical numbers to solve optimization problems?

A: To use critical numbers to solve optimization problems, you need to first find the critical numbers of the function. Then, evaluate the function at each critical number to determine the maximum or minimum value.

Q: What are some common applications of critical numbers?

A: Critical numbers have many applications in fields such as physics, engineering, and economics. They are used to model real-world problems and make predictions about the behavior of systems.

Q: How do I know if a critical number is a maximum or minimum?

A: To determine if a critical number is a maximum or minimum, you need to evaluate the second derivative of the function at the critical number. If the second derivative is positive, the critical number is a minimum. If the second derivative is negative, the critical number is a maximum.

Q: What is the second derivative test?

A: The second derivative test is a test that allows you to determine if a critical number is a maximum or minimum by evaluating the second derivative of the function at the critical number.

Q: How do I use the second derivative test to determine if a critical number is a maximum or minimum?

A: To use the second derivative test, you need to first find the second derivative of the function. Then, evaluate the second derivative at the critical number. If the second derivative is positive, the critical number is a minimum. If the second derivative is negative, the critical number is a maximum.

Q: What are some common mistakes to avoid when working with critical numbers?

A: Some common mistakes to avoid when working with critical numbers include:

Not checking if the derivative is equal to zero or undefined at the critical number.
Not evaluating the second derivative at the critical number.
Not considering the domain of the function when finding critical numbers.

Q: How do I know if a critical number is in the domain of the function?

A: To determine if a critical number is in the domain of the function, you need to check if the function is defined at the critical number. If the function is not defined at the critical number, it is not a critical number.

Q: What are some common applications of critical numbers in real-world problems?

A: Critical numbers have many applications in real-world problems, including:

Modeling population growth and decline.
Modeling the spread of diseases.
Modeling the behavior of physical systems.
Modeling the behavior of economic systems.

Q: How do I use critical numbers to model real-world problems?

A: To use critical numbers to model real-world problems, you need to first identify the variables and parameters of the problem. Then, use the critical numbers to determine the maximum or minimum value of the function.

Q: What are some common challenges when working with critical numbers?

A: Some common challenges when working with critical numbers include:

Finding the critical numbers of a function.
Evaluating the second derivative at the critical number.
Considering the domain of the function when finding critical numbers.

Q: How do I overcome these challenges when working with critical numbers?

A: To overcome these challenges, you need to:

Use the product rule and chain rule of differentiation to find the derivative of the function.
Evaluate the second derivative at the critical number.
Consider the domain of the function when finding critical numbers.

Q: What are some common resources for learning about critical numbers?

A: Some common resources for learning about critical numbers include:

Calculus textbooks.
Online tutorials and videos.
Practice problems and exercises.

Q: How do I practice working with critical numbers?

A: To practice working with critical numbers, you need to:

Work on practice problems and exercises.
Use online resources and tutorials.
Take practice quizzes and exams.

Q: What are some common tips for working with critical numbers?

A: Some common tips for working with critical numbers include:

Use the product rule and chain rule of differentiation to find the derivative of the function.
Evaluate the second derivative at the critical number.
Consider the domain of the function when finding critical numbers.

Q: How do I know if I am ready to work with critical numbers?

A: To know if you are ready to work with critical numbers, you need to:

Have a good understanding of the product rule and chain rule of differentiation.
Be able to evaluate the second derivative at the critical number.
Be able to consider the domain of the function when finding critical numbers.