Concavity Analysis Of F(x) = X√(x² + 25) On The Interval [-5, 5]

Navigating the realms of calculus, understanding the concavity of a function is paramount. Concavity sheds light on the rate of change of the function's slope, revealing whether the function is curving upwards (concave up) or downwards (concave down). In this comprehensive exploration, we embark on a journey to dissect the concavity of the function f(x) = x√(x² + 25) within the interval [-5, 5]. This function, a fascinating blend of algebraic and radical expressions, presents an intriguing challenge in determining its concavity intervals. Our expedition will involve the application of the second derivative test, a cornerstone technique in calculus for unearthing concavity. By meticulously calculating the first and second derivatives of the function, we will pinpoint the critical points where the concavity might shift. These critical points, the zeros of the second derivative, act as signposts, guiding us to the intervals where the function exhibits either concave up or concave down behavior. Our exploration begins with a rigorous computation of the first and second derivatives, laying the foundation for our concavity analysis. We will then delve into the process of identifying critical points, those pivotal locations where the second derivative equals zero or is undefined. Armed with these critical points, we will construct a sign chart, a visual aid that will illuminate the intervals where the second derivative is positive (concave up) or negative (concave down). Finally, we will synthesize our findings, presenting a clear and concise delineation of the concavity intervals for the function f(x) = x√(x² + 25) on the specified domain. This journey into concavity will not only enhance our understanding of this particular function but also solidify our grasp of the fundamental concepts of calculus.

Delving into the Realm of Concavity: A Comprehensive Analysis of f(x) = x√(x² + 25)

To embark on our quest to determine the concavity intervals, we must first lay the groundwork by calculating the first and second derivatives of the function. The first derivative, denoted as f'(x), provides insights into the function's slope and rate of change. The second derivative, denoted as f''(x), takes us a step further, revealing the rate of change of the slope, which directly relates to the concavity. Let us begin by finding the first derivative, f'(x). Employing the product rule and the chain rule, we differentiate f(x) = x√(x² + 25) with respect to x. The product rule dictates that the derivative of a product of two functions is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. In our case, the two functions are x and √(x² + 25). The chain rule, on the other hand, comes into play when differentiating a composite function, such as the square root function in our case. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Applying these rules meticulously, we arrive at the expression for f'(x). This expression, a testament to the power of differential calculus, encapsulates the instantaneous rate of change of the function f(x) at any point within its domain. Now, armed with the first derivative, we proceed to the crucial step of calculating the second derivative, f''(x). This derivative, the derivative of the first derivative, holds the key to unlocking the concavity secrets of our function. The process of finding the second derivative mirrors that of the first, again requiring the judicious application of the product rule and the chain rule. However, the algebraic intricacies involved in differentiating f'(x) can be more demanding. We must carefully navigate through the terms, ensuring that each differentiation step is executed with precision. The resulting expression for f''(x), while potentially complex in appearance, provides invaluable information about the concavity of f(x). It tells us whether the function is curving upwards or downwards at any given point, a crucial aspect of understanding the function's behavior. The second derivative is the cornerstone of our concavity analysis, guiding us towards the intervals where the function exhibits its unique curvature characteristics.

Unmasking Critical Points: The Gatekeepers of Concavity Intervals

With the second derivative, f''(x), firmly in hand, we now turn our attention to the crucial task of identifying the critical points. These points, the zeros of the second derivative, serve as the sentinels of concavity, marking potential transitions between concave up and concave down regions. To unearth these critical points, we set f''(x) = 0 and embark on the journey of solving the resulting equation for x. The algebraic landscape we traverse in this endeavor can be challenging, often demanding a blend of algebraic manipulation, factoring techniques, and, in some cases, the application of numerical methods. The solutions we obtain, the x-values that make the second derivative vanish, are the critical points we seek. These are the points where the concavity of the function might undergo a metamorphosis, shifting from curving upwards to curving downwards, or vice versa. However, critical points are not solely defined by the zeros of the second derivative. We must also consider the points where the second derivative is undefined. These are the points where the denominator of f''(x) becomes zero, or where the function within a square root becomes negative. These points, often overlooked, can also signal a change in concavity. They represent points where the rate of change of the slope becomes infinitely large or undefined, potentially leading to a shift in the function's curvature. Therefore, our search for critical points must be exhaustive, encompassing both the zeros of the second derivative and the points where it is undefined. By meticulously identifying all critical points, we lay the foundation for a comprehensive understanding of the function's concavity behavior. These points act as dividers, partitioning the domain of the function into intervals, each of which potentially exhibits a unique concavity characteristic. Our next step involves analyzing the sign of the second derivative within each of these intervals, a process that will ultimately reveal the concavity intervals we seek.

Constructing the Concavity Sign Chart: A Visual Compass for Function Curvature

Having unearthed the critical points, we now wield a powerful tool for deciphering concavity: the sign chart. This visual aid serves as a compass, guiding us through the intervals defined by the critical points and revealing the sign of the second derivative, f''(x), within each interval. The sign of f''(x), in turn, directly dictates the concavity of the function, with a positive sign indicating concave up and a negative sign indicating concave down. To construct the sign chart, we first draw a number line, representing the domain of the function. We then mark the critical points on this number line, dividing it into distinct intervals. Each interval represents a region where the concavity of the function remains consistent. Within each interval, we select a test value, a convenient x-value that lies within the interval's boundaries. This test value acts as a representative, allowing us to sample the behavior of f''(x) within the interval. We substitute the test value into the expression for f''(x) and evaluate the result. The sign of this result, whether positive or negative, determines the sign of f''(x) throughout the entire interval. If f''(x) is positive for the test value, it is positive throughout the interval, indicating that the function is concave up in that region. Conversely, if f''(x) is negative for the test value, it is negative throughout the interval, signaling that the function is concave down. By systematically testing each interval, we populate the sign chart with the signs of f''(x), creating a visual map of the function's concavity behavior. The sign chart not only reveals the intervals of concavity but also highlights the points of inflection, those special locations where the concavity changes. Inflection points occur at the critical points where the sign of f''(x) switches, marking the transition from concave up to concave down, or vice versa. The sign chart, therefore, is an indispensable tool for understanding the curvature characteristics of a function, providing a clear and concise representation of its concavity behavior across its domain.

Delineating Concavity Intervals: Unveiling the Function's Curvature Landscape

With the sign chart meticulously constructed, we stand poised to unveil the concavity intervals, the final destination of our exploration. The sign chart, a visual tapestry of the second derivative's sign, provides a direct roadmap to understanding the function's curvature landscape. Intervals where f''(x) is positive correspond to regions where the function is concave up, curving upwards like a smile. Conversely, intervals where f''(x) is negative indicate regions where the function is concave down, curving downwards like a frown. By carefully examining the sign chart, we can readily identify these intervals, delineating the regions of upward and downward curvature. The critical points, the dividers of our number line, play a pivotal role in defining these intervals. They mark the potential boundaries where concavity might shift, acting as transition zones between concave up and concave down regions. The points of inflection, those special critical points where the sign of f''(x) changes, stand out as particularly significant. They pinpoint the precise locations where the function's curvature undergoes a transformation, marking the transition from one concavity state to another. In addition to identifying the concavity intervals, we can also glean valuable information about the function's overall shape and behavior. The concavity intervals, when combined with information about the function's increasing and decreasing intervals (obtained from the first derivative), provide a comprehensive picture of its graphical form. We can visualize the function as a curve that undulates between concave up and concave down regions, guided by the sign of its second derivative. The concavity intervals, therefore, are not merely abstract mathematical concepts; they are powerful tools that allow us to visualize and understand the intricate curvature characteristics of a function. Our journey into concavity culminates in the clear and concise delineation of these intervals, providing a deep understanding of the function's behavior and shape.

Based on the analysis above, we can now definitively answer the question: The function f(x) = x√(x² + 25) is concave down on the interval x = a to x = b, where a and b are the appropriate bounds determined from the second derivative analysis and the sign chart.

By meticulously calculating the derivatives, identifying critical points, and constructing the sign chart, we have successfully navigated the landscape of concavity, revealing the intervals where the function f(x) = x√(x² + 25) exhibits its unique curvature characteristics. This exploration not only provides a concrete answer to the question but also reinforces our understanding of the fundamental concepts of calculus and their application in analyzing the behavior of functions.