Consider The Function F ( X ) = 7 X − 28 X F(x)=7x-28\sqrt{x} F ( X ) = 7 X − 28 X ​ , Where X ≥ 0 X \geq 0 X ≥ 0 .1. What Is The Absolute Minimum Of F ( X F(x F ( X ]? If The Absolute Minimum Does Not Exist, Type As: DNE Absolute Minimum = □ \square □ (Ex: 1)2. What Is The

Introduction

In calculus, finding the absolute minimum of a function is a crucial concept that helps us understand the behavior of the function. The absolute minimum of a function is the lowest value that the function can attain. In this article, we will explore the function $f(x)=7x-28\sqrt{x}$, where $x \geq 0$, and find its absolute minimum.

Understanding the Function

The given function is $f(x)=7x-28\sqrt{x}$, where $x \geq 0$. This function involves both a linear term and a square root term. To find the absolute minimum, we need to analyze the behavior of the function as $x$ varies.

Finding the Critical Points

To find the critical points of the function, we need to find the values of $x$ for which the derivative of the function is equal to zero or undefined. Let's find the derivative of the function using the power rule and the chain rule.

$$f'(x) = 7 - \frac{28}{2\sqrt{x}}$$

Simplifying the derivative, we get:

$$f'(x) = 7 - \frac{14}{\sqrt{x}}$$

Now, let's set the derivative equal to zero and solve for $x$.

$$7 - \frac{14}{\sqrt{x}} = 0$$

Multiplying both sides by $\sqrt{x}$, we get:

$$7\sqrt{x} - 14 = 0$$

Adding 14 to both sides, we get:

$$7\sqrt{x} = 14$$

Dividing both sides by 7, we get:

$$\sqrt{x} = 2$$

Squaring both sides, we get:

$$x = 4$$

So, the critical point is $x = 4$.

Analyzing the Behavior of the Function

Now that we have found the critical point, let's analyze the behavior of the function as $x$ varies. We can do this by examining the sign of the derivative in the intervals $x < 4$ and $x > 4$.

For $x < 4$, the derivative is positive, which means that the function is increasing in this interval.

For $x > 4$, the derivative is negative, which means that the function is decreasing in this interval.

Finding the Absolute Minimum

Since the function is increasing for $x < 4$ and decreasing for $x > 4$, the absolute minimum must occur at the critical point $x = 4$.

To find the absolute minimum, we need to evaluate the function at the critical point.

$$f(4) = 7(4) - 28\sqrt{4}$$

Simplifying the expression, we get:

$$f(4) = 28 - 56$$

$$f(4) = -28$$

So, the absolute minimum of the function is $-28$.

Conclusion

In this article, we found the absolute minimum of the function $f(x)=7x-28\sqrt{x}$, where $x \geq 0$. We analyzed the behavior of the function as $x$ varies and found that the absolute minimum occurs at the critical point $x = 4$. The absolute minimum of the function is $-28$.

Absolute Minimum

Absolute minimum = $\boxed{-28}$

Introduction

In our previous article, we explored the function $f(x)=7x-28\sqrt{x}$, where $x \geq 0$, and found its absolute minimum. In this article, we will answer some frequently asked questions related to finding the absolute minimum of a function.

Q: What is the absolute minimum of a function?

A: The absolute minimum of a function is the lowest value that the function can attain. It is the minimum value of the function that occurs at a critical point or at the endpoints of the domain.

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

A: To find the absolute minimum of a function, you need to follow these steps:

1. Find the critical points of the function by setting the derivative equal to zero and solving for $x$.
2. Analyze the behavior of the function as $x$ varies by examining the sign of the derivative in the intervals around the critical points.
3. Evaluate the function at the critical points and at the endpoints of the domain.
4. Compare the values of the function at the critical points and the endpoints to find the absolute minimum.

Q: What is a critical point?

A: A critical point is a value of $x$ for which the derivative of the function is equal to zero or undefined. Critical points are important because they can be the location of the absolute minimum or maximum of the function.

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

A: To determine if a critical point is a minimum or maximum, you need to examine the behavior of the function as $x$ varies around the critical point. If the function is increasing on one side of the critical point and decreasing on the other side, then the critical point is a local maximum. If the function is decreasing on one side of the critical point and increasing on the other side, then the critical point is a local minimum.

Q: What is the difference between a local minimum and an absolute minimum?

A: A local minimum is the minimum value of the function in a small interval around the critical point. An absolute minimum is the minimum value of the function in the entire domain.

Q: Can a function have multiple absolute minima?

A: No, a function can only have one absolute minimum. However, a function can have multiple local minima.

Q: How do I find the absolute minimum of a function with multiple local minima?

A: To find the absolute minimum of a function with multiple local minima, you need to evaluate the function at all the local minima and compare the values to find the absolute minimum.

Q: What if the function has no critical points?

A: If the function has no critical points, then the absolute minimum may occur at the endpoints of the domain. You need to evaluate the function at the endpoints and compare the values to find the absolute minimum.

Conclusion

In this article, we answered some frequently asked questions related to finding the absolute minimum of a function. We hope that this article has helped you to understand the concept of absolute minimum and how to find it. If you have any more questions, please feel free to ask.

Absolute Minimum

Absolute minimum = $\boxed{-28}$