Find The { X$} − V A L U E S O F A L L P O I N T S W H E R E T H E F U N C T I O N H A S A N Y R E L A T I V E E X T R E M A . A L S O , F I N D T H E V A L U E ( S ) O F A N Y R E L A T I V E E X T R E M A . -values Of All Points Where The Function Has Any Relative Extrema. Also, Find The Value(s) Of Any Relative Extrema. − V A L U Eso F A Llp O In T S W H Ere T H E F U N C T I O Nha S An Yre L A T I V Ee X T Re Ma . A L So , F In D T H E V A L U E ( S ) O F An Yre L A T I V Ee X T Re Ma . { F(x) = 3 - (8 + 3x)^{2/3} \}

by ADMIN 527 views

Introduction

Relative extrema are critical points on a function's graph where the function changes from increasing to decreasing or vice versa. These points are essential in understanding the behavior of a function and are used in various applications, including optimization problems and economics. In this article, we will focus on finding the x-values of all points where the function has any relative extrema and determine the value(s) of any relative extrema.

Understanding the Function

The given function is:

f(x)=3−(8+3x)2/3{ f(x) = 3 - (8 + 3x)^{2/3} }

This function involves a cube root and a constant term. To find the relative extrema, we need to find the critical points of the function, which are the points where the derivative of the function is equal to zero or undefined.

Finding the Derivative

To find the derivative of the function, we will use the chain rule. The derivative of the function is:

f′(x)=−23(8+3x)−1/3⋅3{ f'(x) = -\frac{2}{3}(8 + 3x)^{-1/3} \cdot 3 }

Simplifying the derivative, we get:

f′(x)=−2(8+3x)1/3{ f'(x) = -\frac{2}{(8 + 3x)^{1/3}} }

Finding Critical Points

To find the critical points, we need to set the derivative equal to zero and solve for x:

−2(8+3x)1/3=0{ -\frac{2}{(8 + 3x)^{1/3}} = 0 }

Since the denominator cannot be equal to zero, we can multiply both sides by the denominator to get:

−2=0{ -2 = 0 }

This equation has no solution, which means that there are no critical points where the derivative is equal to zero. However, we need to check if the derivative is undefined at any point.

Checking for Undefined Derivative

The derivative is undefined when the denominator is equal to zero:

(8+3x)1/3=0{ (8 + 3x)^{1/3} = 0 }

Solving for x, we get:

8+3x=0{ 8 + 3x = 0 }

3x=−8{ 3x = -8 }

x=−83{ x = -\frac{8}{3} }

This is the only point where the derivative is undefined.

Finding Relative Extrema

Since there are no critical points where the derivative is equal to zero, we need to check the points where the derivative is undefined. In this case, we have only one point:

x=−83{ x = -\frac{8}{3} }

To determine if this point is a relative extremum, we need to check the sign of the derivative on either side of this point.

Checking the Sign of the Derivative

Let's check the sign of the derivative on the left side of the point x = -8/3:

x<−83{ x < -\frac{8}{3} }

Substituting x = -10 into the derivative, we get:

f′(−10)=−2(8+3(−10))1/3{ f'(-10) = -\frac{2}{(8 + 3(-10))^{1/3}} }

f′(−10)=−2(8−30)1/3{ f'(-10) = -\frac{2}{(8 - 30)^{1/3}} }

f′(−10)=−2(−22)1/3{ f'(-10) = -\frac{2}{(-22)^{1/3}} }

f′(−10)=−2−2.5{ f'(-10) = -\frac{2}{-2.5} }

f′(−10)=0.8{ f'(-10) = 0.8 }

Since the derivative is positive on the left side of the point x = -8/3, we can conclude that this point is a relative minimum.

Conclusion

In this article, we found the x-values of all points where the function has any relative extrema and determined the value(s) of any relative extrema. We found that the function has a relative minimum at x = -8/3. This point is critical in understanding the behavior of the function and is used in various applications.

Final Answer

The final answer is:

  • The function has a relative minimum at x = -8/3.
  • The value of the relative minimum is f(-8/3) = 3 - (8 + 3(-8/3))^(2/3) = 3 - (8 - 8)^(2/3) = 3.

Note: The final answer is in the format of a boxed answer, but since the answer is a mathematical expression, it is not possible to put it in a box.

Introduction

Relative extrema are critical points on a function's graph where the function changes from increasing to decreasing or vice versa. In our previous article, we discussed how to find the x-values of all points where the function has any relative extrema and determine the value(s) of any relative extrema. In this article, we will provide a Q&A guide to help you better understand relative extrema and how to find them.

Q: What is a relative extremum?

A: A relative extremum is a critical point on a function's graph where the function changes from increasing to decreasing or vice versa.

Q: How do I find the x-values of all points where the function has any relative extrema?

A: To find the x-values of all points where the function has any relative extrema, you need to find the critical points of the function. Critical points are the points where the derivative of the function is equal to zero or undefined.

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

A: To find the derivative of a function, you can use the power rule, the product rule, and the quotient rule. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). The quotient rule states that if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2.

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

A: To find the critical points of a function, you need to set the derivative equal to zero and solve for x. You also need to check if the derivative is undefined at any point.

Q: What is the difference between a relative minimum and a relative maximum?

A: A relative minimum is a critical point where the function changes from decreasing to increasing, while a relative maximum is a critical point where the function changes from increasing to decreasing.

Q: How do I determine if a critical point is a relative minimum or a relative maximum?

A: To determine if a critical point is a relative minimum or a relative maximum, you need to check the sign of the derivative on either side of the critical point. If the derivative is positive on one side and negative on the other side, then the critical point is a relative extremum.

Q: Can a function have multiple relative extrema?

A: Yes, a function can have multiple relative extrema. For example, a function can have a relative minimum and a relative maximum.

Q: Can a function have no relative extrema?

A: Yes, a function can have no relative extrema. For example, a function that is always increasing or always decreasing has no relative extrema.

Q: How do I use relative extrema in real-world applications?

A: Relative extrema are used in various real-world applications, including optimization problems and economics. For example, a company may want to maximize its profit, which is a relative maximum. A government may want to minimize the cost of a project, which is a relative minimum.

Conclusion

In this article, we provided a Q&A guide to help you better understand relative extrema and how to find them. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of relative extrema.

Final Answer

The final answer is:

  • Relative extrema are critical points on a function's graph where the function changes from increasing to decreasing or vice versa.
  • To find the x-values of all points where the function has any relative extrema, you need to find the critical points of the function.
  • Critical points are the points where the derivative of the function is equal to zero or undefined.
  • To determine if a critical point is a relative minimum or a relative maximum, you need to check the sign of the derivative on either side of the critical point.
  • A function can have multiple relative extrema or no relative extrema.
  • Relative extrema are used in various real-world applications, including optimization problems and economics.