Continuity And Differentiability Analysis Of A Piecewise Function F(x)
In the realm of calculus, understanding the behavior of functions is paramount, especially when dealing with piecewise functions. These functions, defined by different expressions over different intervals, often present intriguing challenges concerning continuity and differentiability. This article delves into a specific piecewise function, analyzing its continuity, first derivative continuity, and second derivative continuity over a given interval. We will rigorously examine the function's definition, apply relevant calculus principles, and arrive at a conclusive determination regarding its properties. Let's consider the function:
f(x) = \begin{cases} \frac{x^2}{2}, & 0 \le x < 1 \\ 2x^2 - 3x + \frac{3}{2}, & 1 \le x \le 2 \end{cases}
The primary objective of this article is to ascertain which of the following statements holds true for this function:
(A) f is continuous in [0, 2]. (B) f' is continuous in [0, 2]. (C) f'' is continuous in [0, 2].
To achieve this, we will systematically investigate the function's behavior at the critical point where the definition changes, which is at x = 1. We will also examine the function's derivatives and their continuity. A thorough understanding of continuity and differentiability is crucial for various applications in mathematics, physics, and engineering. This analysis will provide a comprehensive understanding of the given piecewise function and its properties, emphasizing the importance of rigorous mathematical analysis in determining the behavior of functions. By meticulously examining the function's definition and its derivatives, we can accurately assess its continuity and differentiability across the specified interval, ultimately leading to the correct conclusion among the provided statements. The process involves checking for continuity at the boundary point x=1, calculating the first and second derivatives, and then verifying their continuity at the same point. This step-by-step approach ensures a clear and logical understanding of the function's characteristics.
To determine the continuity of f(x) in the interval [0, 2], we must first verify its continuity at the point where the function's definition changes, which is at x = 1. For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function's value at that point must all be equal. This is the fundamental principle of continuity analysis. In other words, the function should not have any abrupt jumps or breaks at the point in question. We start by calculating the left-hand limit as x approaches 1:
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} \frac{x^2}{2} = \frac{1^2}{2} = \frac{1}{2}
Next, we calculate the right-hand limit as x approaches 1:
\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x^2 - 3x + \frac{3}{2}) = 2(1)^2 - 3(1) + \frac{3}{2} = 2 - 3 + \frac{3}{2} = \frac{1}{2}
Now, we evaluate the function at x = 1:
f(1) = 2(1)^2 - 3(1) + \frac{3}{2} = \frac{1}{2}
Since the left-hand limit, the right-hand limit, and the function's value at x = 1 are all equal to 1/2, we can conclude that f(x) is continuous at x = 1. Additionally, the function is a polynomial and therefore continuous on the interval [0, 1). Similarly, the function is also a polynomial and continuous on the interval (1, 2]. Since f(x) is continuous at x = 1 and continuous on the other intervals as well, we can definitively state that f(x) is continuous on the entire interval [0, 2]. This comprehensive continuity analysis demonstrates that the function does not have any discontinuities within the specified domain, making it a well-behaved function from a continuity perspective. The fact that both parts of the piecewise function are polynomials further simplifies the analysis, as polynomials are inherently continuous. Therefore, the key focus is on the point where the two pieces connect, which we have rigorously verified to be continuous.
Having established the continuity of f(x), we now turn our attention to its differentiability. To analyze the continuity of f'(x) in the interval [0, 2], we first need to compute the derivative of f(x). We differentiate each piece of the piecewise function separately:
For 0 ≤ x < 1:
f'(x) = \frac{d}{dx} (\frac{x^2}{2}) = x
For 1 ≤ x ≤ 2:
f'(x) = \frac{d}{dx} (2x^2 - 3x + \frac{3}{2}) = 4x - 3
Thus, the derivative function f'(x) is given by:
f'(x) = \begin{cases} x, & 0 \le x < 1 \\ 4x - 3, & 1 \le x \le 2 \end{cases}
Now, we need to check the continuity of f'(x) at x = 1. We calculate the left-hand limit, the right-hand limit, and the function's value at x = 1:
Left-hand limit:
\lim_{x \to 1^-} f'(x) = \lim_{x \to 1^-} x = 1
Right-hand limit:
\lim_{x \to 1^+} f'(x) = \lim_{x \to 1^+} (4x - 3) = 4(1) - 3 = 1
Function's value at x = 1:
f'(1) = 4(1) - 3 = 1
Since the left-hand limit, the right-hand limit, and the function's value at x = 1 are all equal to 1, we conclude that f'(x) is continuous at x = 1. Moreover, f'(x) = x is continuous on [0, 1) and f'(x) = 4x - 3 is continuous on (1, 2]. Therefore, f'(x) is continuous on the entire interval [0, 2]. This demonstrates that the first derivative of the function is well-behaved and does not exhibit any discontinuities within the specified domain. The continuity of the first derivative is a crucial aspect in understanding the smoothness of the original function. A continuous first derivative indicates that the function's slope changes gradually, without any abrupt shifts. This differentiability analysis is essential for applications such as optimization and curve sketching, where the behavior of the first derivative provides valuable insights into the function's characteristics.
Having established the continuity of f'(x), we now proceed to analyze the continuity of the second derivative, f''(x). To do this, we first need to compute the second derivative by differentiating f'(x). We differentiate each piece of the piecewise function f'(x) separately:
For 0 ≤ x < 1:
f''(x) = \frac{d}{dx} (x) = 1
For 1 ≤ x ≤ 2:
f''(x) = \frac{d}{dx} (4x - 3) = 4
Thus, the second derivative function f''(x) is given by:
f''(x) = \begin{cases} 1, & 0 \le x < 1 \\ 4, & 1 \le x \le 2 \end{cases}
Now, we need to check the continuity of f''(x) at x = 1. We calculate the left-hand limit and the right-hand limit:
Left-hand limit:
\lim_{x \to 1^-} f''(x) = \lim_{x \to 1^-} 1 = 1
Right-hand limit:
\lim_{x \to 1^+} f''(x) = \lim_{x \to 1^+} 4 = 4
Since the left-hand limit (1) and the right-hand limit (4) are not equal at x = 1, we conclude that f''(x) is discontinuous at x = 1. Although f''(x) is constant on the intervals [0, 1) and (1, 2], the discontinuity at x = 1 means that f''(x) is not continuous on the entire interval [0, 2]. This discontinuity in the second derivative indicates a sudden change in the concavity of the original function at x = 1. The differentiability analysis of the second derivative provides valuable information about the function's curvature and inflection points. A discontinuous second derivative, as we have found in this case, signifies a point where the rate of change of the slope experiences an abrupt jump. This understanding is crucial in various applications, including curve fitting and optimization problems, where the concavity of the function plays a significant role. The discontinuity in f''(x) highlights the importance of meticulously examining the behavior of derivatives at the boundaries of piecewise function definitions.
In summary, through a detailed analysis of the piecewise function f(x), we have determined the following:
- f(x) is continuous on the interval [0, 2].
- f'(x) is continuous on the interval [0, 2].
- f''(x) is discontinuous at x = 1, and therefore not continuous on the interval [0, 2].
Therefore, the correct statements are (A) and (B). Statement (C) is false. This comprehensive analysis underscores the importance of carefully examining the continuity and differentiability of functions, especially piecewise functions, at the points where their definitions change. Understanding these properties is crucial for various applications in mathematics, physics, and engineering. The meticulous examination of limits and function values at critical points allows us to accurately assess the behavior of functions and their derivatives. This, in turn, provides valuable insights into the function's overall characteristics, such as its smoothness, concavity, and points of inflection. By rigorously applying the principles of calculus, we can confidently determine the continuity and differentiability of complex functions, ensuring a solid foundation for further mathematical analysis and problem-solving. The analysis presented here serves as a clear demonstration of how to approach the continuity and differentiability assessment of piecewise functions, emphasizing the need for a systematic and thorough methodology.
