Understanding Absolute Extrema

Oct 13, 2025 by windailys.com 10 views

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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. In this article, we will focus on finding the exact location of all absolute extrema of the function $f(x) = x^3 - 10x$ with domain $[-8, 8]$ .

Understanding Absolute Extrema

Before we dive into the problem, let's understand what absolute extrema are. The absolute extrema of a function are the maximum and minimum values of the function over a given interval. In other words, they are the highest and lowest points of the function within the given interval.

Step 1: Find the Critical Points

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

Finding the Derivative

The derivative of the function $f(x) = x^3 - 10x$ is given by:

f'(x) = 3x^2 - 10

Setting the Derivative Equal to Zero

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

3x^2 - 10 = 0

3x^2 = 10

x^2 = \frac{10}{3}

x = \pm \sqrt{\frac{10}{3}}

Simplifying the Critical Points

The critical points are given by:

x = \pm \sqrt{\frac{10}{3}}

x = \pm \sqrt{\frac{10}{3}} \approx \pm 1.825

Step 2: Find the Second Derivative

To determine whether the critical points correspond to a maximum or minimum, we need to find the second derivative of the function.

Finding the Second Derivative

The second derivative of the function $f(x) = x^3 - 10x$ is given by:

f''(x) = 6x

Step 3: Evaluate the Second Derivative at the Critical Points

Now, we evaluate the second derivative at the critical points:

Evaluating the Second Derivative at $x = \sqrt{\frac{10}{3}}$

f''(\sqrt{\frac{10}{3}}) = 6\sqrt{\frac{10}{3}} > 0

Evaluating the Second Derivative at $x = -\sqrt{\frac{10}{3}}$

f''(-\sqrt{\frac{10}{3}}) = -6\sqrt{\frac{10}{3}} < 0

Step 4: Determine the Nature of the Critical Points

Based on the sign of the second derivative, we can determine the nature of the critical points:

Critical Point $x = \sqrt{\frac{10}{3}}$

Since $f''(\sqrt{\frac{10}{3}}) > 0$ , the critical point $x = \sqrt{\frac{10}{3}}$ corresponds to a minimum.

Critical Point $x = -\sqrt{\frac{10}{3}}$

Since $f''(-\sqrt{\frac{10}{3}}) < 0$ , the critical point $x = -\sqrt{\frac{10}{3}}$ corresponds to a maximum.

Step 5: Find the Absolute Extrema

Now that we have found the critical points and determined their nature, we can find the absolute extrema of the function.

Absolute Minimum

The absolute minimum of the function occurs at the critical point $x = \sqrt{\frac{10}{3}}$ . To find the value of the function at this point, we substitute $x = \sqrt{\frac{10}{3}}$ into the function:

f(\sqrt{\frac{10}{3}}) = (\sqrt{\frac{10}{3}})^3 - 10(\sqrt{\frac{10}{3}})

f(\sqrt{\frac{10}{3}}) = \frac{10\sqrt{10}}{3\sqrt{3}} - \frac{10\sqrt{30}}{3}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\frac{10}{3}}) = -\frac{10\sqrt{30}}{3} + \frac{10\sqrt{10}}{3\sqrt{3}}

f(\sqrt{\<br/> # Finding Absolute Extrema of a Function: A Q&A Guide =========================================================== ## Introduction --------------- In our previous article, we discussed how to find the absolute extrema of a function using calculus. In this article, we will provide a Q&A guide to help you understand the concept better. ## Q: What is the definition of absolute extrema? -------------------------------------------- A: The absolute extrema of a function are the maximum and minimum values of the function over a given interval. ## Q: How do I find the absolute extrema of a function? ----------------------------------------------- A: To find the absolute extrema of a function, you need to follow these steps: 1. Find the critical points of the function by taking the derivative and setting it equal to zero. 2. Determine the nature of the critical points by taking the second derivative and evaluating it at the critical points. 3. Find the absolute minimum and maximum values of the function by evaluating the function at the critical points. ## Q: What is the difference between a local maximum and a global maximum? --------------------------------------------------------- A: A local maximum is a point where the function changes from increasing to decreasing, while a global maximum is the highest point of the function over the entire interval. ## Q: How do I determine the nature of a critical point? ------------------------------------------------ A: To determine the nature of a critical point, you need to take the second derivative and evaluate it at the critical point. If the second derivative is positive, the critical point is a minimum. If the second derivative is negative, the critical point is a maximum. ## Q: What is the significance of the second derivative in finding absolute extrema? --------------------------------------------------------- A: The second derivative is used to determine the nature of the critical points. It helps us to identify whether a critical point is a minimum or a maximum. ## Q: Can a function have multiple absolute extrema? ------------------------------------------------ A: Yes, a function can have multiple absolute extrema. For example, a function can have multiple local maxima and minima, and the global maximum and minimum can occur at different points. ## Q: How do I find the absolute extrema of a function with multiple critical points? --------------------------------------------------------- A: To find the absolute extrema of a function with multiple critical points, you need to follow the same steps as before. However, you need to be careful when evaluating the function at the critical points, as there may be multiple local maxima and minima. ## Q: What is the role of the domain in finding absolute extrema? --------------------------------------------------------- A: The domain of the function is crucial in finding absolute extrema. The domain determines the interval over which the function is defined, and the absolute extrema must occur within this interval. ## Q: Can a function have absolute extrema outside the domain? ------------------------------------------------ A: No, a function cannot have absolute extrema outside the domain. The absolute extrema must occur within the domain of the function. ## Q: How do I apply the concept of absolute extrema in real-world problems? --------------------------------------------------------- A: The concept of absolute extrema is widely used in various fields, such as economics, physics, and engineering. For example, in economics, the absolute extrema of a function can represent the maximum or minimum profit or cost of a business. In physics, the absolute extrema of a function can represent the maximum or minimum energy of a system. ## Q: What are some common applications of absolute extrema? ------------------------------------------------ A: Some common applications of absolute extrema include: * Finding the maximum or minimum profit or cost of a business * Determining the maximum or minimum energy of a system * Finding the maximum or minimum distance between two points * Determining the maximum or minimum speed of an object ## Q: How do I use technology to find absolute extrema? ------------------------------------------------ A: There are various software packages and online tools that can help you find absolute extrema. Some popular options include: * Graphing calculators * Computer algebra systems (CAS) * Online graphing tools * Optimization software ## Q: What are some common mistakes to avoid when finding absolute extrema? --------------------------------------------------------- A: Some common mistakes to avoid when finding absolute extrema include: * Failing to consider the domain of the function * Failing to evaluate the function at all critical points * Failing to determine the nature of the critical points * Failing to consider multiple local maxima and minima ## Q: How do I verify the absolute extrema of a function? ------------------------------------------------ A: To verify the absolute extrema of a function, you need to check that the function is continuous and differentiable over the interval, and that the critical points are the only points where the function changes from increasing to decreasing or from decreasing to increasing. ## Q: What are some common challenges when finding absolute extrema? --------------------------------------------------------- A: Some common challenges when finding absolute extrema include: * Finding the critical points of a function * Determining the nature of the critical points * Evaluating the function at all critical points * Considering multiple local maxima and minima ## Q: How do I overcome these challenges? ------------------------------------------------ A: To overcome these challenges, you need to: * Use technology, such as graphing calculators or computer algebra systems (CAS) * Break down the problem into smaller sub-problems * Use mathematical techniques, such as the second derivative test * Check your work carefully to ensure that you have found the correct absolute extrema.