Find Critical Points And Intervals Of Increase Decrease For F(x) = (1/3)x³ + (17/2)x² + 72x + 8
Introduction
In this article, we will delve into the process of finding the critical points and determining the intervals where the function f(x) = (1/3)x³ + (17/2)x² + 72x + 8 is either increasing or decreasing. This is a fundamental concept in calculus, enabling us to understand the behavior and shape of a function's graph. By identifying critical points, which are the points where the derivative of the function is either zero or undefined, we can pinpoint potential local maxima, local minima, and inflection points. Furthermore, by analyzing the intervals between these critical points, we can determine where the function is increasing (its slope is positive) and where it is decreasing (its slope is negative). This comprehensive analysis provides valuable insights into the function's overall characteristics and behavior.
Understanding the increasing and decreasing nature of a function is crucial in various applications, including optimization problems, curve sketching, and understanding the rate of change of a function. In optimization, we can find the maximum or minimum values of a function by identifying the critical points and analyzing the intervals where the function changes from increasing to decreasing or vice versa. In curve sketching, we can use the information about increasing and decreasing intervals to accurately sketch the graph of a function. In the context of rates of change, we can interpret the intervals of increase and decrease as periods where the function's value is growing or shrinking, respectively. Therefore, mastering this concept is essential for anyone working with calculus and its applications.
We will walk through each step in detail, providing clear explanations and examples to illustrate the concepts involved. Our approach will involve finding the first derivative of the function, setting it equal to zero to find critical points, and then using these critical points to define intervals where we can test the sign of the derivative. A positive derivative indicates that the function is increasing in that interval, while a negative derivative indicates that the function is decreasing. By following this systematic approach, we will gain a comprehensive understanding of how to determine the increasing and decreasing behavior of the given function. This skill is invaluable in various mathematical and real-world applications, making it a crucial topic to master in calculus.
1. Find the First Derivative
The first crucial step in determining the intervals of increase and decrease for a given function is to find its first derivative. The first derivative, denoted as f'(x), provides information about the slope of the tangent line to the function at any given point. This slope directly corresponds to the rate of change of the function; if the derivative is positive, the function is increasing, and if it's negative, the function is decreasing. Understanding and correctly calculating the first derivative is therefore essential for analyzing the behavior of the function.
To find the derivative of f(x) = (1/3)x³ + (17/2)x² + 72x + 8, we will apply the power rule, which states that the derivative of x^n is nx^(n-1). This rule is a cornerstone of differential calculus and allows us to efficiently find the derivatives of polynomial terms. The constant multiple rule, which states that the derivative of cf(x) is cf'(x), where c is a constant, will also be used to handle the coefficients in our function. These rules are fundamental and provide a straightforward way to differentiate polynomial functions.
Applying the power rule and the constant multiple rule to each term of our function, we proceed as follows:
- The derivative of (1/3)x³ is (1/3) * 3x^(3-1) = x².
- The derivative of (17/2)x² is (17/2) * 2x^(2-1) = 17x.
- The derivative of 72x is 72 * 1x^(1-1) = 72.
- The derivative of the constant term 8 is 0, as the derivative of any constant is always zero.
Summing these individual derivatives, we obtain the first derivative of the function:
f'(x) = x² + 17x + 72
This quadratic expression represents the slope of the original function at any point x. The next step involves using this derivative to find the critical points of the function, which will help us identify intervals where the function is increasing or decreasing. Correctly finding the first derivative is a critical step, as any error here will propagate through the rest of the analysis. With the first derivative in hand, we are now well-equipped to proceed with identifying the critical points and analyzing the function's behavior.
2. Find the Critical Points
Now that we have the first derivative, f'(x) = x² + 17x + 72, the next crucial step is to find the critical points of the function. Critical points are the points where the derivative is either equal to zero or undefined. These points are significant because they represent potential locations of local maxima, local minima, or points of inflection on the function's graph. They serve as critical boundaries that divide the domain of the function into intervals where the function's behavior (increasing or decreasing) remains consistent.
In our case, the derivative f'(x) = x² + 17x + 72 is a polynomial, which means it is defined for all real numbers. Therefore, we only need to consider the case where the derivative is equal to zero. To find these points, we set the derivative equal to zero and solve for x:
x² + 17x + 72 = 0
This is a quadratic equation, and we can solve it by factoring. We look for two numbers that multiply to 72 and add to 17. These numbers are 8 and 9. Thus, we can factor the quadratic equation as follows:
(x + 8)(x + 9) = 0
Setting each factor equal to zero gives us the solutions:
x + 8 = 0 => x = -8 x + 9 = 0 => x = -9
Therefore, the critical points of the function are x = -9 and x = -8. These are the x-values where the function's slope could potentially change direction, transitioning from increasing to decreasing or vice versa. It is crucial to correctly identify these critical points, as they form the basis for our subsequent interval analysis. These critical points divide the real number line into three intervals: (-∞, -9), (-9, -8), and (-8, ∞). In the next step, we will analyze the sign of the first derivative in each of these intervals to determine where the function is increasing and decreasing. Finding the critical points accurately is a key step in understanding the overall behavior of the function.
3. Determine Intervals of Increase and Decrease
With the critical points x = -9 and x = -8 identified, we can now determine the intervals where the function is increasing or decreasing. These critical points divide the real number line into three intervals: (-∞, -9), (-9, -8), and (-8, ∞). To determine the behavior of the function within each interval, we will analyze the sign of the first derivative, f'(x) = x² + 17x + 72, in each interval.
To do this, we select a test value within each interval and evaluate the derivative at that point. The sign of the derivative at the test value will indicate whether the function is increasing (positive derivative) or decreasing (negative derivative) throughout the entire interval. This method works because the derivative can only change sign at the critical points, where it equals zero. Therefore, the sign of the derivative remains consistent within each interval defined by the critical points.
Let's analyze each interval:
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Interval (-∞, -9):
- Choose a test value, for example, x = -10.
- Evaluate f'(-10) = (-10)² + 17(-10) + 72 = 100 - 170 + 72 = 2. Since f'(-10) > 0, the function is increasing on the interval (-∞, -9).
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Interval (-9, -8):
- Choose a test value, for example, x = -8.5.
- Evaluate f'(-8.5) = (-8.5)² + 17(-8.5) + 72 = 72.25 - 144.5 + 72 = -0.25. Since f'(-8.5) < 0, the function is decreasing on the interval (-9, -8).
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Interval (-8, ∞):
- Choose a test value, for example, x = 0.
- Evaluate f'(0) = (0)² + 17(0) + 72 = 72. Since f'(0) > 0, the function is increasing on the interval (-8, ∞).
By analyzing the sign of the first derivative in each interval, we have successfully determined where the function f(x) is increasing and decreasing. This information is essential for sketching the graph of the function and understanding its overall behavior. Specifically, we've identified that the function is increasing on the intervals (-∞, -9) and (-8, ∞), and decreasing on the interval (-9, -8). This detailed analysis of intervals of increase and decrease provides valuable insights into the function's characteristics and is a crucial aspect of calculus.
Conclusion
In conclusion, we have successfully identified the critical points and determined the intervals of increase and decrease for the function f(x) = (1/3)x³ + (17/2)x² + 72x + 8. This process involved finding the first derivative, f'(x) = x² + 17x + 72, setting it equal to zero to find the critical points x = -9 and x = -8, and then analyzing the sign of the derivative in the intervals defined by these critical points.
We found that the function is increasing on the intervals (-∞, -9) and (-8, ∞), and decreasing on the interval (-9, -8). This information provides a clear understanding of how the function's values change over its domain. The intervals of increase and decrease are crucial for sketching the graph of the function, identifying local maxima and minima, and solving optimization problems.
The critical points x = -9 and x = -8 are particularly significant. At x = -9, the function changes from increasing to decreasing, indicating a local maximum. Conversely, at x = -8, the function changes from decreasing to increasing, indicating a local minimum. These points are essential landmarks on the function's graph and provide valuable insights into the function's behavior.
By following this systematic approach, we have demonstrated how to analyze the increasing and decreasing behavior of a function using calculus techniques. This method is widely applicable to various functions and is a fundamental skill in calculus. Understanding the intervals of increase and decrease, along with critical points, is crucial for a comprehensive understanding of function behavior and its applications in mathematics and other fields. The process we've outlined serves as a robust framework for analyzing function behavior and can be applied to a wide range of mathematical problems.
