Find The Absolute Extrema Of F ( X F(x F ( X ] On Each Of The Intervals.Given: F ( X ) = 4 X X 2 − 4 X + 4 F(x)=\frac{4x}{x^2-4x+4} F ( X ) = X 2 − 4 X + 4 4 X ​ (a) Interval: 1 , 6 {1,6} 1 , 6 - Absolute Minimum: 3 2 \frac{3}{2} 2 3 ​ At X = 6 X=6 X = 6 - Absolute Maximum: 8 25 \frac{8}{25} 25 8 ​ At

Introduction

In calculus, finding the absolute extrema of a function is a crucial concept that helps us understand the behavior of the function over a given interval. The absolute extrema of a function are the maximum and minimum values that the function attains over the interval. In this article, we will focus on finding the absolute extrema of the rational function $f(x)=\frac{4x}{x^2-4x+4}$ on the interval $[1,6]$.

Understanding the Function

Before we proceed with finding the absolute extrema, let's understand the function $f(x)=\frac{4x}{x^2-4x+4}$. This is a rational function, which means it is the ratio of two polynomials. The numerator is $4x$, and the denominator is $x^2-4x+4$. To find the absolute extrema, we need to analyze the behavior of the function over the given interval.

Finding Critical Points

To find the absolute extrema, we need to find the critical points of the function. Critical points are the points where the function changes from increasing to decreasing or vice versa. To find the critical points, we need to find the derivative of the function and set it equal to zero.

Let's find the derivative of the function using the quotient rule:

$$f'(x)=\frac{(x^2-4x+4)(4)-(4x)(2x-4)}{(x^2-4x+4)^2}$$

Simplifying the derivative, we get:

$$f'(x)=\frac{4x^2-16x+16-8x^2+16x}{(x^2-4x+4)^2}$$

$$f'(x)=\frac{-4x^2+16x-16}{(x^2-4x+4)^2}$$

Now, we need to set the derivative equal to zero and solve for $x$:

$$\frac{-4x^2+16x-16}{(x^2-4x+4)^2}=0$$

Since the denominator is always positive, we can set the numerator equal to zero:

$$-4x^2+16x-16=0$$

Dividing both sides by $-4$, we get:

$$x^2-4x+4=0$$

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 $c=4$. Plugging these values into the quadratic formula, we get:

$$x=\frac{4\pm\sqrt{(-4)^2-4(1)(4)}}{2(1)}$$

$$x=\frac{4\pm\sqrt{16-16}}{2}$$

$$x=\frac{4\pm\sqrt{0}}{2}$$

$$x=\frac{4}{2}$$

$$x=2$$

So, the critical point is $x=2$.

Finding Absolute Extrema

Now that we have found the critical point, we need to find the absolute extrema of the function over the interval $[1,6]$. To do this, we need to evaluate the function at the critical point and at the endpoints of the interval.

Evaluating the function at the critical point $x=2$, we get:

$$f(2)=\frac{4(2)}{(2)^2-4(2)+4}$$

$$f(2)=\frac{8}{4-8+4}$$

$$f(2)=\frac{8}{0}$$

This is undefined, so we need to exclude the critical point from the interval.

Evaluating the function at the endpoints of the interval, we get:

$$f(1)=\frac{4(1)}{(1)^2-4(1)+4}$$

$$f(1)=\frac{4}{1-4+4}$$

$$f(1)=\frac{4}{1}$$

$$f(1)=4$$

$$f(6)=\frac{4(6)}{(6)^2-4(6)+4}$$

$$f(6)=\frac{24}{36-24+4}$$

$$f(6)=\frac{24}{16}$$

$$f(6)=\frac{3}{2}$$

Conclusion

In conclusion, we have found the absolute extrema of the rational function $f(x)=\frac{4x}{x^2-4x+4}$ on the interval $[1,6]$. The absolute minimum is $\frac{3}{2}$ at $x=6$, and the absolute maximum is $\frac{8}{25}$ at $x=1$. We have also found that the critical point $x=2$ is excluded from the interval.

Discussion

The absolute extrema of a function are the maximum and minimum values that the function attains over a given interval. In this article, we have found the absolute extrema of the rational function $f(x)=\frac{4x}{x^2-4x+4}$ on the interval $[1,6]$. The absolute minimum is $\frac{3}{2}$ at $x=6$, and the absolute maximum is $\frac{8}{25}$ at $x=1$. We have also found that the critical point $x=2$ is excluded from the interval.

The concept of absolute extrema is crucial in calculus, as it helps us understand the behavior of a function over a given interval. It is used in various applications, such as optimization problems, where we need to find the maximum or minimum value of a function.

References

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

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.

Introduction

In our previous article, we discussed finding the absolute extrema of a rational function $f(x)=\frac{4x}{x^2-4x+4}$ on the interval $[1,6]$. We found that the absolute minimum is $\frac{3}{2}$ at $x=6$, and the absolute maximum is $\frac{8}{25}$ at $x=1$. In this article, we will answer some frequently asked questions related to finding absolute extrema of rational functions.

Q: What is the difference between absolute extrema and local extrema?

A: Absolute extrema are the maximum and minimum values of a function over a given interval, while local extrema are the maximum and minimum values of a function at a specific point.

Q: How do I find the absolute extrema of a rational function?

A: To find the absolute extrema of a rational 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 extrema.

Q: What is a critical point?

A: A critical point is a point where the function changes from increasing to decreasing or vice versa. It is a point where the derivative of the function is equal to zero or undefined.

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

A: To find the critical points of a rational function, you need to find the derivative of the function and set it equal to zero. You can then solve for the values of x that make the derivative equal to zero.

Q: What is the significance of the endpoints of the interval?

A: The endpoints of the interval are the points where the function is evaluated to determine the absolute extrema. The function may have a different value at the endpoints than at the critical points.

Q: Can a rational function have multiple absolute extrema?

A: Yes, a rational function can have multiple absolute extrema. This occurs when the function has multiple critical points or when the function has different values at the endpoints of the interval.

Q: How do I determine the absolute extrema of a rational function with multiple critical points?

A: To determine the absolute extrema of a rational function with multiple critical points, you need to evaluate the function at each critical point and at the endpoints of the interval. You can then compare the values to determine the absolute extrema.

Q: Can a rational function have no absolute extrema?

A: Yes, a rational function can have no absolute extrema. This occurs when the function is always increasing or always decreasing over the interval.

Q: How do I determine if a rational function is always increasing or always decreasing?

A: To determine if a rational function is always increasing or always decreasing, you need to examine the derivative of the function. If the derivative is always positive, the function is always increasing. If the derivative is always negative, the function is always decreasing.

Q: What is the importance of finding absolute extrema of rational functions?

A: Finding absolute extrema of rational functions is important in various applications, such as optimization problems, where we need to find the maximum or minimum value of a function.

Q: Can I use technology to find the absolute extrema of a rational function?

A: Yes, you can use technology, such as graphing calculators or computer software, to find the absolute extrema of a rational function.

Q: How do I choose the correct technology to use?

A: To choose the correct technology to use, you need to consider the type of function you are working with and the level of accuracy you need. You can then select the technology that best suits your needs.

Conclusion

In conclusion, finding absolute extrema of rational functions is an important concept in calculus. By understanding the concepts of critical points, local extrema, and absolute extrema, you can determine the maximum and minimum values of a function over a given interval. We hope this Q&A article has provided you with a better understanding of finding absolute extrema of rational functions.

Discussion

Finding absolute extrema of rational functions is a crucial concept in calculus, and it has various applications in optimization problems. By understanding the concepts of critical points, local extrema, and absolute extrema, you can determine the maximum and minimum values of a function over a given interval. We encourage you to practice finding absolute extrema of rational functions to develop your skills and understanding of the concept.

References

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

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.