Derivatives And Intervals Of Increase For F(x) = 5x² - 4x - 6

Hey everyone! Today, we're diving into the world of calculus with a fun little function: f(x) = 5x² - 4x - 6. We're going to explore its derivatives and figure out where this function is increasing. So, buckle up and let's get started!

(a) Finding the First Derivative, f'(x)

Let's kick things off by finding the first derivative, f'(x). We've got two main ways to tackle this: using the definition of a derivative or using the power rule. For this one, let's use the power rule since it's a bit quicker and cleaner. The power rule states that if we have a term axⁿ, its derivative is naxⁿ⁻¹. So, we'll apply this to each term in our function. The first derivative is a fundamental concept in calculus, representing the instantaneous rate of change of a function. It essentially tells us how the function's output changes with respect to its input at any given point. Finding the first derivative can be approached in two primary ways: using the definition of a derivative (a limit-based approach) or applying established derivative rules (such as the power rule, sum/difference rule, etc.). The definition of the derivative provides a foundational understanding, while the rules offer a more efficient method for many common functions. Understanding the first derivative is crucial for various applications, including optimization problems, finding critical points, and analyzing the behavior of a function. For example, setting the first derivative to zero helps identify points where the function reaches a local maximum or minimum. Furthermore, the sign of the first derivative indicates whether the function is increasing or decreasing over a given interval. So, mastering the techniques for finding and interpreting the first derivative is essential for anyone delving into calculus and its applications. The power rule is a cornerstone of differential calculus, providing a direct method for finding the derivatives of polynomial terms. Its simplicity and efficiency make it a preferred tool for handling functions involving powers of x. However, it's important to remember that the power rule is just one of several derivative rules, each designed to handle different types of functions and operations. For more complex functions, a combination of rules may be required. For instance, the product rule and quotient rule are essential for differentiating functions that are products or quotients of other functions. Similarly, the chain rule is crucial for composite functions. Mastering these rules, alongside the power rule, allows for the differentiation of a wide range of functions. Ultimately, the goal is to develop a strong understanding of the underlying principles and techniques so that you can confidently tackle derivative problems, regardless of their complexity.

• Term 1: 5x². Applying the power rule, we get 2 * 5x¹ = 10x.
• Term 2: -4x. This is the same as -4x¹, so the derivative is 1 * -4x⁰ = -4 (remember, anything to the power of 0 is 1).
• Term 3: -6. This is a constant, and the derivative of a constant is always 0.

So, putting it all together, f'(x) = 10x - 4. Boom! We found the first derivative.

(b) Finding the Second Derivative, f''(x)

Now, let's find the second derivative, f''(x). This is simply the derivative of the first derivative. So, we'll take f'(x) = 10x - 4 and differentiate it again. The second derivative builds upon the concept of the first derivative by providing information about the rate of change of the first derivative itself. In simpler terms, it tells us how the slope of the function is changing. A positive second derivative indicates that the slope is increasing, resulting in a concave up shape, while a negative second derivative means the slope is decreasing, leading to a concave down shape. Finding the second derivative involves differentiating the first derivative, which may require applying the same derivative rules as before, such as the power rule, sum/difference rule, and so on. The second derivative has significant applications in various areas of calculus and related fields. For instance, it plays a crucial role in optimization problems, particularly in determining whether a critical point is a local maximum, a local minimum, or a point of inflection. By examining the sign of the second derivative at a critical point, we can classify the nature of that point. Moreover, the second derivative is used in physics to describe acceleration, which is the rate of change of velocity. Understanding the second derivative enhances our ability to analyze the behavior of functions and their graphs. By considering both the first and second derivatives, we gain a comprehensive picture of a function's characteristics, including its increasing/decreasing intervals, concavity, and points of inflection. This knowledge is invaluable for sketching accurate graphs of functions and solving a wide range of problems in calculus and beyond. The concavity of a function, indicated by the sign of its second derivative, provides essential information about its curvature. A concave up section resembles a smile, while a concave down section resembles a frown. Points of inflection, where the concavity changes, are of particular interest as they mark significant transitions in the function's behavior. Analyzing concavity and points of inflection helps us understand how the function's rate of change is itself changing. This is particularly useful in applications like optimization, where identifying the maximum or minimum values of a function requires understanding its shape and curvature. Furthermore, in fields like economics and engineering, concavity can represent concepts such as diminishing returns or the stability of a system. Therefore, the second derivative is not merely a mathematical tool but a powerful concept for interpreting and modeling real-world phenomena.

• Term 1: 10x. The derivative is simply 10.
• Term 2: -4. This is a constant, so its derivative is 0.

Therefore, f''(x) = 10. Nice and easy!

(c) Determining the Interval Where f is Increasing

Okay, time to figure out where our function f(x) is increasing. Remember, a function is increasing where its first derivative, f'(x), is positive. So, we need to solve the inequality f'(x) > 0. To determine the intervals where a function is increasing, we focus on analyzing its first derivative, f'(x). The fundamental principle is that a function is increasing over an interval where its derivative is positive. This is because a positive derivative indicates that the function's slope is upward, meaning its values are increasing as x increases. The process of finding these intervals typically involves several steps. First, we find the first derivative of the function using appropriate differentiation techniques. Next, we identify the critical points of the function, which are the points where the first derivative is either zero or undefined. These critical points are crucial because they mark potential turning points where the function may switch from increasing to decreasing or vice versa. Once we have the critical points, we divide the domain of the function into intervals based on these points. Then, we test the sign of the first derivative within each interval. This can be done by choosing a test value within the interval and evaluating the derivative at that point. If the derivative is positive, the function is increasing in that interval; if it's negative, the function is decreasing. The intervals of increasing and decreasing are key characteristics of a function, providing insights into its behavior and shape. This analysis is not only essential for understanding the function itself but also for various applications, such as optimization problems and curve sketching. In optimization, for example, identifying intervals of increasing and decreasing helps pinpoint local maxima and minima. Similarly, in curve sketching, this information aids in creating an accurate representation of the function's graph. The relationship between the first derivative and the increasing/decreasing behavior of a function is a cornerstone of calculus, providing a powerful tool for analyzing and understanding functions. This concept extends to more advanced topics, such as the analysis of higher-order derivatives and their relationship to concavity and other function properties. Therefore, mastering the techniques for finding and interpreting intervals of increasing and decreasing is fundamental for anyone studying calculus and its applications. Remember, the goal is to connect the mathematical concepts with the visual representation of the function, allowing for a deeper understanding of its behavior.

We found that f'(x) = 10x - 4. So, let's solve 10x - 4 > 0.

1. Add 4 to both sides: 10x > 4
2. Divide both sides by 10: x > 4/10
3. Simplify: x > 2/5

So, f(x) is increasing on the interval (2/5, ∞).

Wrapping Up

And there you have it! We successfully found the first and second derivatives of f(x) = 5x² - 4x - 6 and determined the interval where it's increasing. Hope this was helpful and fun! Keep exploring the wonderful world of calculus, guys!