Which Of The Two Functions Below Has The Smallest Minimum { Y $} − V A L U E ? -value? − V A L U E ? { \begin{array}{c} f(x) = (x-13)^4 - 2 \\ g(x) = 3x^3 + 2 \end{array} \} A. There Is Not Enough Information To Determine. B. { F(x) $}$ C.

When comparing the minimum values of two functions, it's essential to understand the characteristics of each function and how they behave as the input variable changes. In this article, we'll compare the minimum values of two given functions, $f(x) = (x-13)^4 - 2$ and $g(x) = 3x^3 + 2$, to determine which one has the smallest minimum $y$-value.

Understanding the Functions

Function $f(x)$

The function $f(x) = (x-13)^4 - 2$ is a quartic function, which means it has a degree of 4. This function has a unique characteristic: it has a single minimum point, which occurs when the expression inside the parentheses is equal to zero. In other words, the minimum point occurs when $x-13=0$, which gives us $x=13$. The value of the function at this point is $f(13) = (13-13)^4 - 2 = -2$.

Function $g(x)$

The function $g(x) = 3x^3 + 2$ is a cubic function, which means it has a degree of 3. This function has a unique characteristic: it has a single inflection point, which occurs when the derivative of the function is equal to zero. In other words, the inflection point occurs when $g'(x) = 9x^2 = 0$, which gives us $x=0$. The value of the function at this point is $g(0) = 3(0)^3 + 2 = 2$.

Comparing the Minimum Values

Now that we have a good understanding of both functions, let's compare their minimum values. We know that the minimum value of $f(x)$ occurs at $x=13$, and the value of the function at this point is $f(13) = -2$. We also know that the minimum value of $g(x)$ occurs at $x=0$, and the value of the function at this point is $g(0) = 2$.

Since $-2 < 2$, we can conclude that the minimum value of $f(x)$ is smaller than the minimum value of $g(x)$.

Conclusion

In conclusion, the function $f(x) = (x-13)^4 - 2$ has the smallest minimum $y$-value compared to the function $g(x) = 3x^3 + 2$. This is because the minimum value of $f(x)$ occurs at $x=13$, and the value of the function at this point is $f(13) = -2$, which is smaller than the minimum value of $g(x)$.

Discussion

This problem requires a good understanding of the characteristics of different types of functions, including quartic and cubic functions. It also requires the ability to analyze and compare the minimum values of two functions. In this case, we were able to determine that the minimum value of $f(x)$ is smaller than the minimum value of $g(x)$ by analyzing the behavior of each function and comparing their minimum values.

Key Takeaways

• The minimum value of a function occurs at a point where the derivative of the function is equal to zero.
• The minimum value of a quartic function occurs at a single point, while the minimum value of a cubic function occurs at a single inflection point.
• To compare the minimum values of two functions, we need to analyze the behavior of each function and compare their minimum values.

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition.
• [2] Algebra and Trigonometry, Michael Sullivan, 4th edition.

Additional Resources

• Khan Academy: Calculus I, II, and III
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Calculus and Algebra

Conclusion

In our previous article, we compared the minimum values of two functions, $f(x) = (x-13)^4 - 2$ and $g(x) = 3x^3 + 2$, to determine which one has the smallest minimum $y$-value. In this article, we'll answer some frequently asked questions related to this topic.

Q: What is the minimum value of a function?

A: The minimum value of a function is the smallest value that the function can take. It occurs at a point where the derivative of the function is equal to zero.

Q: How do you find the minimum value of a function?

A: To find the minimum value of a function, you need to find the point where the derivative of the function is equal to zero. This point is called the critical point. You can then evaluate the function at this point to find the minimum value.

Q: What is the difference between a minimum value and an inflection point?

A: A minimum value is the smallest value that a function can take, while an inflection point is a point where the function changes from concave up to concave down or vice versa. An inflection point is not necessarily a minimum value.

Q: Can a function have multiple minimum values?

A: No, a function can only have one minimum value. However, a function can have multiple inflection points.

Q: How do you compare the minimum values of two functions?

A: To compare the minimum values of two functions, you need to find the minimum value of each function and then compare them. You can do this by evaluating the functions at their critical points or by using calculus techniques such as optimization.

Q: What is the significance of the minimum value of a function?

A: The minimum value of a function is significant because it represents the smallest value that the function can take. It is often used in optimization problems, where the goal is to minimize a function.

Q: Can you give an example of a function with a minimum value?

A: Yes, the function $f(x) = (x-13)^4 - 2$ has a minimum value of -2, which occurs at $x=13$.

Q: Can you give an example of a function with an inflection point but no minimum value?

A: Yes, the function $g(x) = 3x^3 + 2$ has an inflection point at $x=0$, but it does not have a minimum value.

Q: How do you determine if a function has a minimum value or an inflection point?

A: To determine if a function has a minimum value or an inflection point, you need to analyze the function's behavior and look for points where the derivative is equal to zero. If the derivative is equal to zero, it may indicate a minimum value or an inflection point.

Q: Can you use calculus techniques to find the minimum value of a function?

A: Yes, you can use calculus techniques such as optimization to find the minimum value of a function. Optimization involves finding the maximum or minimum value of a function subject to certain constraints.

Q: What are some common applications of minimum values?

A: Minimum values are used in a variety of applications, including optimization problems, economics, and engineering. They are often used to minimize costs, maximize profits, or optimize systems.

Conclusion

In conclusion, the minimum value of a function is a critical concept in calculus and optimization. It represents the smallest value that a function can take and is often used in a variety of applications. By understanding the minimum value of a function, you can better analyze and compare the behavior of different functions.