Finding The Inflection Point Of F(x) = X³ + 21x² + 18x + 2

To find the inflection point of the function f(x) = x³ + 21x² + 18x + 2, we need to determine where the concavity of the function changes. This involves finding the second derivative of the function and setting it equal to zero. Let's break down the process step-by-step. Inflection points are crucial in calculus as they indicate a change in the curvature of a graph, transitioning from concave up to concave down, or vice versa. Understanding how to find these points is fundamental in analyzing the behavior of functions and their graphical representations. This article provides a comprehensive guide on locating inflection points, using the given function as a practical example. We will explore the necessary calculus techniques, ensuring that readers gain a thorough understanding of the concept and the methods involved.

Step 1: Find the First Derivative

The first derivative, denoted as f'(x), gives us the rate of change of the function. To find it, we apply the power rule to each term in the original function:

f(x) = x³ + 21x² + 18x + 2

Applying the power rule (d/dx (xⁿ) = nxⁿ⁻¹), we get:

f'(x) = 3x² + 42x + 18

The first derivative represents the slope of the tangent line at any point on the function’s graph. It is a vital tool for identifying critical points, which include local maxima, minima, and saddle points. By setting the first derivative to zero and solving for x, we can find these critical points. However, for inflection points, we need to go a step further and examine the second derivative. The first derivative not only helps in finding critical points but also provides insights into the function's increasing and decreasing intervals. Regions where f'(x) > 0 indicate an increasing function, while regions where f'(x) < 0 indicate a decreasing function. This information is invaluable for sketching the graph of the function and understanding its overall behavior.

Step 2: Find the Second Derivative

The second derivative, denoted as f''(x), gives us the rate of change of the first derivative and indicates the concavity of the function. Again, we apply the power rule to f'(x):

f'(x) = 3x² + 42x + 18

Differentiating this, we get:

f''(x) = 6x + 42

The second derivative is the key to identifying inflection points. It tells us how the slope of the tangent line is changing. When f''(x) > 0, the function is concave up, meaning it curves upwards like a smile. Conversely, when f''(x) < 0, the function is concave down, curving downwards like a frown. Inflection points occur where the concavity changes, and this happens where f''(x) = 0 or is undefined. The second derivative test is a powerful tool for determining the nature of critical points as well. If a critical point c satisfies f''(c) > 0, then f(x) has a local minimum at c. If f''(c) < 0, then f(x) has a local maximum at c. If f''(c) = 0, the test is inconclusive, and further analysis is required.

Step 3: Set the Second Derivative Equal to Zero and Solve for x

To find potential inflection points, we set f''(x) equal to zero:

6x + 42 = 0

Now, solve for x:

6x = -42

x = -7

This x value is a potential inflection point. However, we need to confirm that the concavity changes at this point. To ensure that x = -7 corresponds to an inflection point, we must verify that the second derivative changes sign around this value. This is typically done by testing intervals to the left and right of x = -7. The value of x = -7 is a critical candidate for an inflection point, but it is not conclusive evidence on its own. The sign change in the second derivative is the definitive indicator of an inflection point. If the second derivative does not change sign, then the point is merely a point of horizontal tangency in the concavity, but not an inflection point.

Step 4: Determine the y-coordinate

To find the corresponding y-coordinate, we plug x = -7 back into the original function:

f(-7) = (-7)³ + 21(-7)² + 18(-7) + 2

f(-7) = -343 + 1029 - 126 + 2

f(-7) = 562

So, the potential inflection point is (-7, 562). Finding the y-coordinate is a crucial step in fully defining the inflection point. It provides the exact location of the point on the graph where the concavity changes. This coordinate, along with the x-coordinate, gives a complete picture of where the function's curvature transitions. Without the y-coordinate, we would only know the x-value at which the inflection might occur, but not its precise location on the graph. This step ensures that we have a tangible point that can be plotted and visually verified on the function's curve.

Step 5: Verify the Inflection Point

To confirm that (-7, 562) is indeed an inflection point, we need to check the sign of f''(x) on either side of x = -7. Let's test x = -8 and x = -6:

For x = -8:

f''(-8) = 6(-8) + 42 = -48 + 42 = -6

Since f''(-8) < 0, the function is concave down to the left of x = -7.

For x = -6:

f''(-6) = 6(-6) + 42 = -36 + 42 = 6

Since f''(-6) > 0, the function is concave up to the right of x = -7.

Because the concavity changes at x = -7, the point (-7, 562) is indeed an inflection point. Verification is a critical step in the process of finding inflection points. It ensures that the potential inflection point identified by setting the second derivative to zero is actually a point where the concavity changes. By testing values on either side of the x-coordinate, we can confirm whether the second derivative changes sign, which is the defining characteristic of an inflection point. This step eliminates the possibility of false positives and provides a rigorous confirmation of the inflection point's existence. The change in concavity has significant implications for the shape of the graph and the behavior of the function.

Conclusion

The inflection point for the function f(x) = x³ + 21x² + 18x + 2 is (-7, 562). This point marks the transition where the function changes from concave down to concave up. Inflection points are significant features of a function’s graph, providing valuable insights into its behavior and curvature. Understanding how to find them is essential in calculus and mathematical analysis. The process of finding inflection points involves differentiation, setting derivatives to zero, and verifying the change in concavity. This comprehensive approach ensures accurate identification of these critical points. Inflection points, along with other critical points such as maxima and minima, help in creating a complete picture of a function's characteristics and its graphical representation. By mastering the techniques for finding these points, students and professionals alike can gain a deeper understanding of mathematical functions and their applications in various fields.

In summary, finding inflection points is a systematic process involving the first and second derivatives. It is a fundamental concept in calculus with broad applications in mathematics, physics, engineering, and economics. The ability to accurately determine inflection points enhances our understanding of function behavior and graphical representation.