Analyzing F(x) = 5x + 3x^-1 Intervals And Critical Points
Introduction
In the realm of calculus, understanding the behavior of functions is paramount. This involves identifying critical points, determining intervals of increase and decrease, and analyzing concavity. In this article, we delve into a comprehensive analysis of the function f(x) = 5x + 3x^-1. We will pinpoint critical points, delineate intervals of interest, and discuss the function's behavior across these intervals. Understanding these aspects is crucial for sketching the graph of the function and comprehending its overall characteristics. We aim to provide a clear, step-by-step analysis suitable for students and enthusiasts alike. The heart of calculus lies in understanding functions, and this exploration serves as a valuable exercise in that pursuit. This function, while seemingly simple, exhibits interesting behavior that warrants a detailed examination. By exploring its derivatives, critical points, and intervals, we can gain a deeper appreciation for the interplay between algebraic expressions and graphical representations.
Finding Critical Points
To begin our analysis, we must first identify the critical points of the function f(x) = 5x + 3x^-1. Critical points are crucial because they often indicate local maxima, local minima, or points of inflection. They are the turning points of the function's graph, where the slope changes direction. To find these points, we need to calculate the first derivative of f(x) and set it equal to zero. The derivative, denoted as f'(x), represents the instantaneous rate of change of the function. The process of finding critical points is fundamental in calculus and provides essential information about the function's behavior. Let's embark on this process. We start by rewriting the function as f(x) = 5x + 3/x to make differentiation easier. Now, we apply the power rule and the constant multiple rule to find the derivative.
f'(x) = 5 - 3x^-2
This derivative tells us how the function's slope changes with respect to x. To find the critical points, we set f'(x) equal to zero and solve for x. This will give us the x-values where the tangent line to the function's graph is horizontal. These are the points where the function potentially changes from increasing to decreasing or vice versa. The algebraic manipulation to solve for x is straightforward but crucial. It involves isolating the x term and taking the square root. Remember, when taking the square root, we must consider both positive and negative solutions. This is because both values, when squared, will yield the same positive result. The critical points are key features of the function's graph, acting as potential maxima, minima, or inflection points. Their identification is a crucial step in our analysis.
Setting f'(x) = 0, we get:
5 - 3x^-2 = 0
5 = 3/x^2
5x^2 = 3
x^2 = 3/5
x = ±√(3/5)
Thus, the critical points are x = -√(3/5) and x = √(3/5). These points, approximately -0.7746 and 0.7746, divide the number line into intervals where the function's behavior is consistent. We will use these critical points to define the intervals and analyze whether the function is increasing or decreasing in each interval.
Defining Intervals
The critical points we found, x = -√(3/5) and x = √(3/5), are pivotal in defining the intervals where the function's behavior remains consistent. These points act as dividers on the number line, creating segments where the function is either increasing or decreasing. We must also consider the point where the function is undefined, which is x = 0 due to the term 3x^-1 in the original function. This point adds another layer to our interval analysis. The inclusion of x = 0 is crucial because it represents a vertical asymptote, where the function's value approaches infinity or negative infinity. Ignoring this point would lead to an incomplete and potentially misleading analysis of the function's behavior. The intervals we define will help us understand how the function changes and behaves across its entire domain. This is essential for sketching an accurate graph and understanding the function's overall properties. By carefully considering the critical points and points of discontinuity, we can create a comprehensive picture of the function's behavior.
Therefore, we have four important intervals to consider:
- (-∞, -√(3/5)]
- [-√(3/5), 0)
- (0, √(3/5)]
- [√(3/5), ∞)
These intervals will be the focus of our subsequent analysis. We will determine whether the function is increasing or decreasing in each of these intervals by examining the sign of the first derivative. This will provide us with a clear understanding of the function's overall behavior and its graphical representation. The next step is to test the sign of the first derivative within each interval. This will tell us whether the function is increasing or decreasing in that interval. We will choose a test value within each interval, plug it into the first derivative, and observe the sign of the result. This process will allow us to create a sign chart, which visually summarizes the function's increasing and decreasing behavior across its domain.
Analyzing Intervals of Increase and Decrease
Now, we will analyze the intervals of increase and decrease for the function f(x) = 5x + 3x^-1. This involves determining where the function's graph is sloping upwards (increasing) and where it is sloping downwards (decreasing). The key to this analysis lies in examining the sign of the first derivative, f'(x) = 5 - 3x^-2, within each of the intervals we defined earlier. A positive f'(x) indicates an increasing function, while a negative f'(x) indicates a decreasing function. This is a fundamental concept in calculus, allowing us to connect the algebraic expression of the derivative to the graphical behavior of the function. By systematically testing the sign of f'(x) in each interval, we can construct a clear picture of the function's increasing and decreasing behavior. This information is crucial for sketching an accurate graph and understanding the function's overall characteristics. The process involves choosing a test value within each interval, plugging it into f'(x), and observing the resulting sign. This might seem like a repetitive process, but it is a reliable method for determining the function's behavior across its domain.
We will choose a test value within each interval and plug it into f'(x) to determine its sign:
-
(-∞, -√(3/5)]: Let's choose x = -1. Then,
f'(-1) = 5 - 3(-1)^-2 = 5 - 3 = 2 > 0
So, the function is increasing in this interval.
-
[-√(3/5), 0): Let's choose x = -0.5. Then,
f'(-0.5) = 5 - 3(-0.5)^-2 = 5 - 3(4) = 5 - 12 = -7 < 0
So, the function is decreasing in this interval.
-
(0, √(3/5)]: Let's choose x = 0.5. Then,
f'(0.5) = 5 - 3(0.5)^-2 = 5 - 3(4) = 5 - 12 = -7 < 0
So, the function is decreasing in this interval.
-
[√(3/5), ∞): Let's choose x = 1. Then,
f'(1) = 5 - 3(1)^-2 = 5 - 3 = 2 > 0
So, the function is increasing in this interval.
This analysis reveals the function's dynamic behavior across its domain. The function increases as we approach the critical point from negative infinity, then decreases until it reaches 0, decreases again until it gets to the positive critical point, and finally increases toward infinity. These behaviors are essential pieces of information when we sketch the function's graph. The sign of the first derivative provides a direct link between the function's algebraic expression and its graphical representation. By understanding this link, we can effectively analyze and interpret the behavior of various functions. Our findings here lay the groundwork for a deeper understanding of the function's local extrema (maxima and minima) and its overall shape.
Identifying Local Extrema
Having determined the intervals of increase and decrease, we can now identify the local extrema of the function f(x) = 5x + 3x^-1. Local extrema are the points where the function reaches a local maximum or a local minimum value. These points are significant because they represent the