Which Of The Two Functions Below Has The Largest Maximum $y$-value?$\[ \begin{align*} f(x) &= -3x^4 - 14 \\ g(x) &= -x^3 + 2 \end{align*} \\]A. $f(x)$ B. The Extreme Maximum $y$-value For Both

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Introduction

When comparing two functions, it's essential to determine which one has the largest maximum y-value. This is crucial in various mathematical applications, such as optimization problems and data analysis. In this article, we will compare the maximum y-values of two given functions, f(x) and g(x), and determine which one has the largest maximum y-value.

The Functions

We are given two functions:

f(x)=−3x4−14{ f(x) = -3x^4 - 14 }

g(x)=−x3+2{ g(x) = -x^3 + 2 }

Understanding the Functions

To compare the maximum y-values of these functions, we need to understand their behavior. The function f(x) is a quartic function, which means it has a degree of 4. This function has a negative leading coefficient, indicating that it opens downwards. The function g(x) is a cubic function, which means it has a degree of 3. This function also has a negative leading coefficient, indicating that it opens downwards.

Finding the Maximum y-Value

To find the maximum y-value of a function, we need to find the critical points of the function. Critical points occur when the derivative of the function is equal to zero or undefined. We will find the critical points of both functions and compare their y-values.

Finding the Critical Points of f(x)

To find the critical points of f(x), we need to find the derivative of f(x) with respect to x.

f′(x)=−12x3{ f'(x) = -12x^3 }

Setting the derivative equal to zero, we get:

−12x3=0{ -12x^3 = 0 }

Solving for x, we get:

x=0{ x = 0 }

This is the only critical point of f(x).

Finding the Critical Points of g(x)

To find the critical points of g(x), we need to find the derivative of g(x) with respect to x.

g′(x)=−3x2{ g'(x) = -3x^2 }

Setting the derivative equal to zero, we get:

−3x2=0{ -3x^2 = 0 }

Solving for x, we get:

x=0{ x = 0 }

This is the only critical point of g(x).

Comparing the y-Values of the Critical Points

Now that we have found the critical points of both functions, we need to compare their y-values. We will substitute the critical points into both functions and compare the results.

For f(x), we substitute x = 0 into the function:

f(0)=−3(0)4−14{ f(0) = -3(0)^4 - 14 }

f(0)=−14{ f(0) = -14 }

For g(x), we substitute x = 0 into the function:

g(0)=−03+2{ g(0) = -0^3 + 2 }

g(0)=2{ g(0) = 2 }

Comparing the y-values, we see that g(0) = 2 is greater than f(0) = -14.

Conclusion

In conclusion, the function g(x) has the largest maximum y-value compared to f(x). This is because g(0) = 2 is greater than f(0) = -14. Therefore, the correct answer is B. The extreme maximum y-value for both.

Discussion

This problem requires a deep understanding of functions and their behavior. It's essential to analyze the functions carefully and find the critical points to determine the maximum y-value. This problem can be extended to more complex functions and applications, such as optimization problems and data analysis.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Functions, 2nd edition, James Stewart

Additional Resources

  • Khan Academy: Functions and Graphs
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Function Analysis
    Q&A: Comparing the Maximum y-Values of Two Functions =====================================================

Introduction

In our previous article, we compared the maximum y-values of two functions, f(x) and g(x), and determined that g(x) has the largest maximum y-value. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the significance of finding the maximum y-value of a function?

A: Finding the maximum y-value of a function is crucial in various mathematical applications, such as optimization problems and data analysis. It helps us understand the behavior of the function and make informed decisions.

Q: How do I find the maximum y-value of a function?

A: To find the maximum y-value of a function, you need to find the critical points of the function. Critical points occur when the derivative of the function is equal to zero or undefined. You can use calculus to find the derivative of the function and set it equal to zero to find the critical points.

Q: What is the difference between a maximum and a minimum y-value?

A: A maximum y-value is the largest value that a function can take, while a minimum y-value is the smallest value that a function can take. In the case of the two functions f(x) and g(x), g(x) has the largest maximum y-value, while f(x) has the smallest minimum y-value.

Q: Can I use other methods to find the maximum y-value of a function?

A: Yes, you can use other methods to find the maximum y-value of a function, such as graphing the function or using numerical methods. However, calculus is often the most efficient and accurate method.

Q: How do I determine which function has the largest maximum y-value?

A: To determine which function has the largest maximum y-value, you need to compare the y-values of the critical points of both functions. You can substitute the critical points into both functions and compare the results.

Q: What are some common mistakes to avoid when comparing the maximum y-values of two functions?

A: Some common mistakes to avoid when comparing the maximum y-values of two functions include:

  • Not finding the critical points of both functions
  • Not comparing the y-values of the critical points
  • Not considering the behavior of the functions at the endpoints of the domain

Q: Can I apply this concept to more complex functions?

A: Yes, you can apply this concept to more complex functions. However, you may need to use more advanced calculus techniques, such as implicit differentiation or parametric differentiation.

Q: What are some real-world applications of comparing the maximum y-values of two functions?

A: Some real-world applications of comparing the maximum y-values of two functions include:

  • Optimization problems, such as finding the maximum profit or minimum cost
  • Data analysis, such as finding the maximum or minimum value of a dataset
  • Physics and engineering, such as finding the maximum or minimum velocity or acceleration of an object

Conclusion

In conclusion, comparing the maximum y-values of two functions is a crucial concept in mathematics and has many real-world applications. By understanding how to find the maximum y-value of a function and comparing it to other functions, you can make informed decisions and solve complex problems.

Discussion

This article provides a comprehensive overview of the concept of comparing the maximum y-values of two functions. It answers frequently asked questions and provides examples and real-world applications to illustrate the concept.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Functions, 2nd edition, James Stewart
  • [3] Khan Academy: Functions and Graphs
  • [4] MIT OpenCourseWare: Calculus
  • [5] Wolfram Alpha: Function Analysis

Additional Resources

  • Khan Academy: Optimization Problems
  • MIT OpenCourseWare: Data Analysis
  • Wolfram Alpha: Physics and Engineering Applications