Finding Critical Numbers And Verifying With Graphing Utility For F(x) = X^2 - 10x

In the realm of calculus, understanding the behavior of functions is paramount. One crucial aspect of this understanding lies in identifying critical numbers, points where the function's derivative is either zero or undefined. These critical numbers serve as potential locations for local maxima, local minima, or saddle points, which are essential features in sketching the graph of a function and analyzing its properties. This article delves into the process of finding critical numbers for the quadratic function $f(x) = x^2 - 10x$, and it will show you how to confirm these findings using a graphing utility. We will explore the concepts of derivatives, critical points, and how they relate to the graphical representation of the function. Our aim is to provide a comprehensive understanding of how to analyze functions using both analytical and graphical methods, which are fundamental skills in calculus and related fields.

(a) Finding the Critical Numbers of f(x) = x^2 - 10x

To determine the critical numbers of the function $f(x) = x^2 - 10x$, we first need to find its derivative, $f'(x)$. The derivative represents the instantaneous rate of change of the function at any given point. Using the power rule of differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$, we can find the derivative of $f(x)$. The power rule is a cornerstone of differential calculus, providing a straightforward method for finding the derivatives of polynomial functions. Applying this rule to our function, we differentiate each term separately. The derivative of $x^2$ is $2x$, and the derivative of $-10x$ is $-10$. Therefore, the derivative of $f(x)$ is:

$f'(x) = 2x - 10$

Critical numbers occur where the derivative is either equal to zero or undefined. In this case, $f'(x)$ is a linear function, which is defined for all real numbers. Thus, we only need to find where $f'(x) = 0$. Setting the derivative equal to zero, we have:

$2x - 10 = 0$

To solve for $x$, we add 10 to both sides of the equation:

$2x = 10$

Then, we divide both sides by 2:

$x = 5$

Therefore, the critical number of the function $f(x) = x^2 - 10x$ is $x = 5$. This single critical number suggests that the function has either a local minimum or a local maximum at this point. To determine which, we would typically use the first or second derivative test. The first derivative test involves analyzing the sign of the derivative on either side of the critical point, while the second derivative test involves evaluating the second derivative at the critical point. In the next sections, we will use a graphing utility to confirm this result and visually analyze the function's behavior around this critical point. Understanding how to find and interpret critical numbers is crucial for analyzing the behavior of functions and solving optimization problems in calculus.

(b) Using a Graphing Utility to Confirm the Critical Number

Now that we have analytically found the critical number of the function $f(x) = x^2 - 10x$ to be $x = 5$, it is essential to verify this result graphically. Graphing utilities, such as Desmos, GeoGebra, or even a handheld graphing calculator, provide a visual representation of the function, allowing us to confirm our analytical findings. Graphing utilities are invaluable tools in calculus, bridging the gap between abstract mathematical concepts and their visual representations. To confirm our critical number, we will graph the function and look for points where the slope of the tangent line is zero, which corresponds to the critical points.

First, input the function $f(x) = x^2 - 10x$ into the graphing utility. The graph will appear as a parabola opening upwards. This shape is characteristic of quadratic functions with a positive leading coefficient. The critical point we found, $x = 5$, should correspond to the vertex of this parabola. The vertex is the point where the parabola changes direction, and it represents either the minimum or maximum value of the function. In this case, since the parabola opens upwards, the vertex represents the minimum value of the function.

Observe the graph closely around the point $x = 5$. You should notice that the parabola reaches its lowest point at this $x$-value. The tangent line at this point is horizontal, indicating that the derivative is zero. This visual confirmation reinforces our analytical result that $x = 5$ is indeed a critical number. To further verify, you can use the graphing utility to find the coordinates of the vertex. Most graphing utilities have features that allow you to find the minimum or maximum point of a function. When you use this feature, you should find that the vertex is at the point $(5, -25)$. This means that the function has a minimum value of -25 at $x = 5$, further confirming our critical number.

The use of a graphing utility not only confirms our analytical solution but also provides a deeper understanding of the function's behavior. It allows us to visualize the relationship between the critical number and the shape of the graph. This visual confirmation is an essential step in problem-solving, ensuring that our analytical calculations are accurate and that we have a comprehensive understanding of the function's properties. In the next section, we will further explore the implications of this critical number and its relationship to the function's increasing and decreasing intervals.

(c) Analyzing Increasing and Decreasing Intervals using the Graph

Having found the critical number $x = 5$ for the function $f(x) = x^2 - 10x$ and confirmed it graphically, we can now analyze the intervals where the function is increasing and decreasing. The critical number divides the domain of the function into intervals, and the sign of the derivative in each interval indicates whether the function is increasing or decreasing. Analyzing intervals is crucial for understanding the overall behavior of a function, including its trends and extrema. A function is increasing where its derivative is positive and decreasing where its derivative is negative. The critical numbers are the potential points where the function changes from increasing to decreasing or vice versa.

Recall that the derivative of $f(x) = x^2 - 10x$ is $f'(x) = 2x - 10$. We found that $f'(x) = 0$ at $x = 5$. Now, we need to determine the sign of $f'(x)$ in the intervals $(-\infty, 5)$ and $(5, \infty)$.

Let's consider the interval $(-\infty, 5)$. Choose a test value in this interval, say $x = 0$. Evaluate $f'(0) = 2(0) - 10 = -10$. Since $f'(0) < 0$, the function is decreasing in the interval $(-\infty, 5)$. This means that as $x$ increases from negative infinity to 5, the value of $f(x)$ decreases.

Now, let's consider the interval $(5, \infty)$. Choose a test value in this interval, say $x = 6$. Evaluate $f'(6) = 2(6) - 10 = 2$. Since $f'(6) > 0$, the function is increasing in the interval $(5, \infty)$. This means that as $x$ increases from 5 to infinity, the value of $f(x)$ increases.

Graphically, we can observe this behavior by looking at the parabola. To the left of $x = 5$, the parabola slopes downwards, indicating that the function is decreasing. To the right of $x = 5$, the parabola slopes upwards, indicating that the function is increasing. This visual representation confirms our analytical findings. The critical point $x = 5$ is a local minimum because the function changes from decreasing to increasing at this point.

Using a graphing utility, you can trace the function and observe its behavior in these intervals. As you move along the graph from left to right, you will see the function decreasing until it reaches the vertex at $x = 5$, and then it starts increasing. This exercise reinforces the relationship between the derivative, the critical numbers, and the increasing/decreasing intervals of a function. Understanding these concepts is fundamental for analyzing the behavior of functions and solving optimization problems.

In conclusion, we have successfully found the critical number of the function $f(x) = x^2 - 10x$ using analytical methods and confirmed our results graphically using a graphing utility. We determined that the critical number is $x = 5$, and this corresponds to a local minimum of the function. Furthermore, we analyzed the increasing and decreasing intervals of the function, finding that it is decreasing on the interval $(-\infty, 5)$ and increasing on the interval $(5, \infty)$.

This process highlights the importance of combining analytical and graphical techniques in calculus. Analytical methods provide precise solutions, while graphical methods offer visual confirmation and a deeper understanding of the function's behavior. The use of a graphing utility not only validates our calculations but also enhances our intuition about the function's properties.

The concepts explored in this article, such as critical numbers, derivatives, increasing and decreasing intervals, are fundamental to calculus and have wide-ranging applications in various fields. Understanding how to find and interpret these features is crucial for solving optimization problems, modeling real-world phenomena, and making informed decisions based on mathematical analysis. By mastering these techniques, students and practitioners can gain a profound understanding of the behavior of functions and their applications in diverse contexts. The ability to analyze functions both analytically and graphically is a valuable skill that empowers individuals to tackle complex problems and gain insights into the mathematical world around us.