Find The Absolute Maximum And Minimum Values Of F ( X ) = 7 − X 3 F(x) = 7 - X^3 F ( X ) = 7 − X 3 On The Interval − 3 ≤ X ≤ 6 -3 \leq X \leq 6 − 3 ≤ X ≤ 6 . Then Graph The Function. Label The Points On The Graph Where The Absolute Extrema Occur, And Include Their Coordinates.- Absolute

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Introduction

In calculus, finding the absolute maximum and minimum values of a function on a given interval is a crucial concept. It involves identifying the points where the function reaches its highest and lowest values within the specified interval. In this article, we will explore how to find the absolute maximum and minimum values of the function $f(x) = 7 - x^3$ on the interval $-3 \leq x \leq 6$. We will also graph the function and label the points where the absolute extrema occur.

Understanding the Function

The given function is $f(x) = 7 - x^3$. This is a cubic function, which means it has a polynomial of degree three. The graph of this function will be a curve that opens downward, as the coefficient of the $x^3$ term is negative.

Finding Critical Points

To find the absolute maximum and minimum values of the function, we need to find the critical points. Critical points are the points where the derivative of the function is equal to zero or undefined. In this case, we need to find the derivative of the function and set it equal to zero.

Derivative of the Function

To find the derivative of the function, we will use the power rule of differentiation. The derivative of $x^n$ is $nx^{n-1}$. Applying this rule to the function $f(x) = 7 - x^3$, we get:

$$f'(x) = -3x^2$$

Setting the Derivative Equal to Zero

Now that we have the derivative of the function, we need to set it equal to zero and solve for $x$. This will give us the critical points.

$$-3x^2 = 0$$

Solving for $x$, we get:

$$x^2 = 0$$

$$x = 0$$

So, the only critical point is $x = 0$.

Finding the Absolute Maximum and Minimum Values

Now that we have the critical point, we need to find the absolute maximum and minimum values of the function. To do this, we will evaluate the function at the critical point and at the endpoints of the interval.

Evaluating the Function at the Critical Point

We will evaluate the function at the critical point $x = 0$.

$$f(0) = 7 - (0)^3$$

$$f(0) = 7$$

So, the value of the function at the critical point is $7$.

Evaluating the Function at the Endpoints

We will evaluate the function at the endpoints of the interval $-3$ and $6$.

$$f(-3) = 7 - (-3)^3$$

$$f(-3) = 7 + 27$$

$$f(-3) = 34$$

$$f(6) = 7 - (6)^3$$

$$f(6) = 7 - 216$$

$$f(6) = -209$$

So, the values of the function at the endpoints are $34$ and $-209$.

Comparing the Values

Now that we have the values of the function at the critical point and at the endpoints, we can compare them to find the absolute maximum and minimum values.

The value of the function at the critical point is $7$. The values of the function at the endpoints are $34$ and $-209$. Since $7$ is greater than $-209$ and less than $34$, the absolute maximum value of the function is $34$ and the absolute minimum value is $-209$.

Graphing the Function

To visualize the absolute maximum and minimum values, we will graph the function.

The graph of the function $f(x) = 7 - x^3$ is a curve that opens downward. The absolute maximum value of the function occurs at the point $(6, 34)$, and the absolute minimum value occurs at the point $(-3, -209)$.

Conclusion

In this article, we found the absolute maximum and minimum values of the function $f(x) = 7 - x^3$ on the interval $-3 \leq x \leq 6$. We also graphed the function and labeled the points where the absolute extrema occur. The absolute maximum value of the function is $34$, which occurs at the point $(6, 34)$, and the absolute minimum value is $-209$, which occurs at the point $(-3, -209)$.

Absolute Extrema

Point	Value
$(6, 34)$	$34$
$(-3, -209)$	$-209$

Discussion

The concept of absolute maximum and minimum values is crucial in calculus. It helps us understand the behavior of functions and identify the points where the function reaches its highest and lowest values. In this article, we applied this concept to the function $f(x) = 7 - x^3$ on the interval $-3 \leq x \leq 6$. We found the absolute maximum and minimum values of the function and graphed the function to visualize the results.

References

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

Introduction

In our previous article, we explored how to find the absolute maximum and minimum values of the function $f(x) = 7 - x^3$ on the interval $-3 \leq x \leq 6$. We also graphed the function and labeled the points where the absolute extrema occur. In this article, we will answer some frequently asked questions related to finding absolute maximum and minimum values.

Q&A

Q: What is the difference between absolute maximum and minimum values?

A: The absolute maximum value of a function is the largest value that the function attains on a given interval, while the absolute minimum value is the smallest value that the function attains on the same interval.

Q: How do I find the absolute maximum and minimum values of a function?

A: To find the absolute maximum and minimum values of a function, you need to find the critical points of the function, evaluate the function at the critical points and at the endpoints of the interval, and compare the values to determine the absolute maximum and minimum values.

Q: What is a critical point?

A: A critical point is a point where the derivative of the function is equal to zero or undefined. Critical points are important because they can be local maxima or minima, or they can be points where the function changes from increasing to decreasing or vice versa.

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you need to apply the power rule of differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$. You can also use other rules of differentiation, such as the product rule and the quotient rule.

Q: What is the significance of the absolute maximum and minimum values?

A: The absolute maximum and minimum values of a function are important because they help us understand the behavior of the function and identify the points where the function reaches its highest and lowest values. This information can be useful in a variety of applications, such as optimization problems and data analysis.

Q: Can I use technology to find the absolute maximum and minimum values of a function?

A: Yes, you can use technology, such as graphing calculators or computer software, to find the absolute maximum and minimum values of a function. These tools can help you visualize the function and identify the points where the absolute extrema occur.

Q: What are some common mistakes to avoid when finding absolute maximum and minimum values?

A: Some common mistakes to avoid when finding absolute maximum and minimum values include:

• Failing to find all critical points
• Failing to evaluate the function at the endpoints of the interval
• Failing to compare the values of the function at the critical points and endpoints
• Using incorrect rules of differentiation

Conclusion

In this article, we answered some frequently asked questions related to finding absolute maximum and minimum values. We hope that this information will be helpful to you as you work on problems involving absolute extrema.

Additional Resources

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Online graphing calculator: www.desmos.com
• [4] Computer software: www.mathematica.com

Discussion

Finding absolute maximum and minimum values is an important concept in calculus. It helps us understand the behavior of functions and identify the points where the function reaches its highest and lowest values. In this article, we provided some answers to frequently asked questions related to finding absolute maximum and minimum values. We hope that this information will be helpful to you as you work on problems involving absolute extrema.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Online graphing calculator: www.desmos.com
• [4] Computer software: www.mathematica.com