Analyzing Function Behavior Critical Points, Intervals, And Extrema For F'(x) = X(x-5)

In calculus, understanding the behavior of a function is crucial. We can gain insights into a function's characteristics, such as its critical points, intervals of increase and decrease, and local extrema, by analyzing its derivative. This article delves into a detailed analysis of a function whose derivative is given by f'(x) = x(x-5). We will explore how to determine the critical points, identify intervals where the function is increasing or decreasing, and locate any local maximum or minimum points. This comprehensive exploration will provide a solid understanding of how derivatives can be used to sketch the graph of a function and understand its behavior.

a. Identifying the Critical Points of f

Critical points, in the realm of calculus, are pivotal points in understanding the behavior of a function. These are the points where the derivative of the function, f'(x), is either equal to zero or undefined. These points are critical because they represent potential locations where the function may change its direction – from increasing to decreasing or vice versa. At these points, the function could potentially reach a local maximum or a local minimum, making them essential for optimization problems and graph sketching.

To determine the critical points of our function, where f'(x) = x(x-5), we need to find the values of x for which f'(x) = 0. This involves setting the derivative equal to zero and solving for x. In this case, we have the equation x(x-5) = 0. This equation is already factored, making it straightforward to find the solutions. The solutions are the values of x that make each factor equal to zero. Thus, we have two possible solutions: x = 0 and x - 5 = 0. Solving the second equation gives us x = 5. Therefore, the critical points of the function f(x) occur at x = 0 and x = 5. These are the points where the tangent line to the graph of f(x) is horizontal, indicating a possible change in the function's direction.

Finding the critical points is just the first step. Once we have these points, we can use them to divide the domain of the function into intervals. These intervals are then analyzed to determine where the function is increasing, decreasing, or constant. The critical points serve as boundaries for these intervals, providing a framework for understanding the overall behavior of the function. By examining the sign of the derivative within each interval, we can determine the function's trend and identify potential local extrema. This process is fundamental to sketching the graph of the function and solving optimization problems.

b. Determining Intervals of Increase and Decrease for f

Understanding where a function is increasing or decreasing is fundamental to grasping its overall behavior. The first derivative, f'(x), provides invaluable information in this regard. If f'(x) > 0 on an interval, it signifies that the function f(x) is increasing over that interval. Conversely, if f'(x) < 0 on an interval, the function f(x) is decreasing. Where f'(x) = 0, as we discussed earlier, we find critical points, which are potential turning points where the function changes its direction.

To determine the intervals of increase and decrease for our function, we use the critical points we found in part (a), x = 0 and x = 5, to divide the number line into intervals. These critical points act as dividers, creating intervals where the sign of f'(x) remains constant. The intervals we need to consider are (-∞, 0), (0, 5), and (5, ∞). Within each of these intervals, we will choose a test value and evaluate f'(x) at that value. The sign of f'(x) at the test value will tell us whether the function is increasing or decreasing throughout the entire interval. This is because the sign of the derivative can only change at the critical points.

Let's take a closer look at each interval. For the interval (-∞, 0), we can choose a test value of x = -1. Evaluating f'(-1) = (-1)(-1-5) = (-1)(-6) = 6, which is positive. This indicates that f(x) is increasing on the interval (-∞, 0). Next, consider the interval (0, 5). A suitable test value would be x = 1. Evaluating f'(1) = (1)(1-5) = (1)(-4) = -4, which is negative. Therefore, f(x) is decreasing on the interval (0, 5). Finally, for the interval (5, ∞), let's use x = 6 as our test value. Evaluating f'(6) = (6)(6-5) = (6)(1) = 6, which is positive. This tells us that f(x) is increasing on the interval (5, ∞). By systematically analyzing the sign of the derivative in each interval, we can accurately map out the intervals where the function is increasing and decreasing. This information is crucial for sketching the graph of the function and identifying its local extrema.

c. Locating Points of Local Extrema of f

Local extrema are critical points where a function reaches a local maximum or a local minimum. These points are essential for understanding the shape of a function's graph and for optimization problems. A local maximum occurs at a point where the function's value is greater than or equal to the values at all nearby points, while a local minimum occurs where the function's value is less than or equal to the values at all nearby points. To identify local extrema, we can use the first derivative test, which relies on the sign changes of the derivative around the critical points.

The First Derivative Test states that if f'(x) changes from positive to negative at a critical point c, then f(x) has a local maximum at x = c. Conversely, if f'(x) changes from negative to positive at x = c, then f(x) has a local minimum at x = c. If f'(x) does not change sign at x = c, then f(x) has neither a local maximum nor a local minimum at that point. This test allows us to analyze the behavior of the function around its critical points and determine the nature of the extrema.

In our case, we have critical points at x = 0 and x = 5. From our analysis in part (b), we know that f'(x) changes from positive to negative at x = 0. This indicates that f(x) has a local maximum at x = 0. To find the value of the local maximum, we would need to evaluate the original function f(x) at x = 0. However, since we are only given the derivative f'(x), we cannot determine the exact value of f(0) without additional information, such as an initial condition or the antiderivative of f'(x).

Similarly, we know that f'(x) changes from negative to positive at x = 5. This indicates that f(x) has a local minimum at x = 5. Again, to find the specific value of the local minimum, we would need to evaluate the original function f(x) at x = 5. Without knowing the original function, we can only conclude that there is a local minimum at x = 5. By applying the first derivative test and analyzing the sign changes of the derivative, we can effectively identify and classify the local extrema of a function. This is a powerful tool in calculus for understanding the behavior of functions and solving optimization problems.

In conclusion, by analyzing the derivative f'(x) = x(x-5), we have successfully identified the critical points of the function, determined the intervals where the function is increasing and decreasing, and located the points where the function has local extrema. The critical points are at x = 0 and x = 5. The function is increasing on the intervals (-∞, 0) and (5, ∞) and decreasing on the interval (0, 5). There is a local maximum at x = 0 and a local minimum at x = 5. This analysis demonstrates the power of calculus in understanding the behavior of functions and provides a comprehensive approach to problem-solving in this area.